Modeling of Physical and Electrical Characteristics of ... · Modeling of Physical and Electrical Characteristics of Organic Thin Film Transistors ... Dept. of Electrical and Electronic

Ph.D. in Electronic and Computer EngineeringDept. of Electrical and Electronic Engineering

University of Cagliari

Modeling of Physical and ElectricalCharacteristics of Organic Thin Film

Transistors

Simone Locci

Advisor : Prof. Dr. Annalisa BonfiglioCurriculum: ING-INF/01 Electronics

XXI CycleFebruary 2009

Ph.D. in Electronic and Computer EngineeringDept. of Electrical and Electronic Engineering

University of Cagliari

Modeling of the Physical andElectrical Characteristics of Organic

Thin Film Transistors

Simone Locci

Advisor : Prof. Dr. Annalisa BonfiglioCurriculum: ING-INF/01 Electronics

XXI CycleFebruary 2009

Contents

Abstract (Italian) ix

1 Introduction 1

2 From charge transport to organic transistors 32.1 Charge transport in conjugated polymers . . . . . . . . . . . . . . . . . . . 3

2.1.1 Hopping between localized states . . . . . . . . . . . . . . . . . . . 52.1.2 Multiple trapping and release model . . . . . . . . . . . . . . . . . 62.1.3 The polaron model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Organic field effect transistors . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Analytical derivation of the current . . . . . . . . . . . . . . . . . . 122.4 Mobility models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4.1 Multiple trap and release mobility . . . . . . . . . . . . . . . . . . . 162.4.2 Grain Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.3 Variable range hopping . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.4 Poole-Frenkel mobility . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Drift diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Cylindrical thin film transistors 273.1 Device structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Linear region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Saturation region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Short channel effects 434.1 The device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.2 Model derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

i

ii CONTENTS

4.2.1 Linear region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.2 Depletion region . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2.3 SCLC region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2.4 The limit of thick film . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Experimental results and parameter extraction . . . . . . . . . . . . . . . . 574.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Dynamic models for organic TFTs 615.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1.1 An arbitrary density of trapped states . . . . . . . . . . . . . . . . 645.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.1 Capacitance-Voltage curves . . . . . . . . . . . . . . . . . . . . . . 675.2.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Conclusion 79

Bibliography 81

List of Publications Related to the Thesis 89

Acknowledgements 91*

List of Figures

2.1 sp2 hybridization of two carbon atoms. sp2 orbitals lie on the same planeand bond into a σ-bond, pz orbitals are orthogonal to the plane and bondinto a π-bond. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Bonding of pz orbitals: depending on the sign of the wavefunctions, π

orbitals with lower energy or π∗ orbitals with higher energy can be originated. 5

2.3 Distribution of trap states in the band gap. Shallow traps energy is justfew kT over the valence band, for a p-type semiconductor (conduction bandfor n-type). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 p-type organic semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.5 n-type organic semiconductors. . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Ambipolar organic semiconductors: (a) oligothiophene/ fullerene dyad andoligothiophene/fullerene triad; (b) 9-(1,3- dithiol-2-ylidene)thioxanthene-C60 system (n = 6); (c) poly(3,9-di-tert-butylindeno[1,2-b]fluorene) (PIF);(d) the near-infrared absorbing dye bis[4-dimethylaminodithiobenzyl]nickel(nickel di- thiolene); (e) quinoidal terthiophene (DCMT). . . . . . . . . . . 11

2.7 Top contact transistor structure. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.8 Bottom contact transistor structure. . . . . . . . . . . . . . . . . . . . . . 12

2.9 Geometry of the bottom contact transistor. . . . . . . . . . . . . . . . . . . 13

2.10 Band diagram for a Schottky contact between an n-type semiconductorand metal at the thermal equilibrium. . . . . . . . . . . . . . . . . . . . . 24

3.1 Geometry of the cylindrical TFT. . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Normalized threshold voltage vs. gate radius for Vfb = 0 V, and ds = 50 nm

and different insulator thicknesses. The curves are normalized with respectto the threshold voltages of planar devices with the same thicknesses. . . . 31

3.3 Depletion region vs. insulator thickness. A planar device and a cylindricaldevice, but with rg = 1 µm, are shown. −Vg + V (z) = 10 V, Vfb = 0. . . . . 34

iii

iv LIST OF FIGURES

3.4 Width of the depletion region vs. gate radius for −Vg + V (z) = 10 V,Vfb = 0 V, di = 500 nm. The depletion region for a planar TFT with thesame insulator characteristics is equal to W = 32.15 nm. . . . . . . . . . . 35

3.5 Section of the TFT transistor. The depletion region at the lower rightcorner has a width W (z). The accumulation layer extends up to za. . . . . 35

3.6 Id − Vd characteristics simulated for cylindrical devices with rg = 1 µm

and rg = 10 µm and a planar device. Both have Vfb = 0 V, di = 500 nm,ds = 20 nm. It is supposed that the devices have Z/L = 1. Characteristicsare simulated for a variation of the gate voltage from 0 V to −100 V insteps of −20 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.7 Id − Vg characteristics for devices with gold and PEDOT contacts. Thedrain voltage is kept at Vd = −100 V. . . . . . . . . . . . . . . . . . . . . . 38

3.8 Mobility vs. gate voltage. The drain voltage is kept at Vd = −100 V. . . . 393.9 Id − Vd characteristic for device with gold contacts. The gate voltage is

kept at Vg = −100 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.10 Id − Vd characteristic for device with PEDOT contacts. The gate voltage

is kept at Vg = −100 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 Geometry of a bottom contact OTFT. . . . . . . . . . . . . . . . . . . . . 444.2 Modes of operation of the OTFT. The origin of the x-axis is at the source

contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.3 Geometry of the thin film n+− i−n+ structure: the device has a length of

l = 4.8 µm and it has a strip contact geometry, with two 25 µm long ohmiccontacts. The structure is surrounded by dv = 10 µm of vacuum. . . . . . 48

4.4 I − V curves for n+ − i− n+ structure for different γ values. . . . . . . . 494.5 Absolute value of the electric field in the structure and in vacuum. . . . . . 504.6 Zoom of the absolute value of the electric field in the structure and in

vacuum. The structure begins at x = 25 µm and ends at x = 29.8 µm. . . . 514.7 Pinch-off abscissa xp as a function of the drain voltage Vd with L = 15 µm,

γ = 1.5 · 10−4 (m/V)1/2 and Vth = 0. Vg = −10,−30, . . . ,−90 V. ForVg = −90 V the device remains in accumulation regime, since Vd < Vg, soxp = L. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.8 Output characteristics for a p-device with L = 15 µm, γ = 1.5·10−3 (m/V)1/2

and Vth = 0. Vg = −10,−30, . . . ,−90 V. . . . . . . . . . . . . . . . . . . . . 534.9 Output characteristics for a p-device with L = 15 µm, γ = 1.5·10−4 (m/V)1/2

and Vth = 0. Vg = −10,−30, . . . ,−90 V. . . . . . . . . . . . . . . . . . . . . 544.10 Output characteristics for p-devices with L = 3, 5, . . . , 13 µm. γ = 1.5 ·

10−4 (m/V)1/2. Vg = −25 V. . . . . . . . . . . . . . . . . . . . . . . . . . . 55

LIST OF FIGURES v

4.11 Id−Vg characteristics for L = 5 µm and L = 2.5 µm devices. Drain voltageis kept at Vd = −75 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.12 Id − Vd characteristics for L = 2.5 µm device. Vg = 0,−20, . . . ,−80 V. . . . 594.13 Id − Vd characteristics for L = 5 µm device. Vg = 0,−20, . . . ,−80 V. . . . . 594.14 Zero-field mobility as a function of the gate voltage. . . . . . . . . . . . . . 60

5.1 Emission and recombination of carriers for a single level of trap. . . . . . . 625.2 Geometry of a bottom contact OTFT. . . . . . . . . . . . . . . . . . . . . 665.3 Geometry of a MIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.4 Equivalent circuit for the capacitance in a MIS structure. . . . . . . . . . . 685.5 C − V simulation for MIS structure with no trap states in the band gap. . 705.6 Id − Vg simulation for OTFT structure with no traps states in the band gap. 705.7 C − V simulation for acceptor traps with densities Ntr = 1011÷ 1013 cm−2.

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 715.8 Id−Vg simulation for acceptor traps with densities Ntr = 1011÷1013 cm−2.

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 715.9 C−V simulation for acceptor traps with σvth = 7.8·10−18÷7.8·10−20 cm3/s.

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 725.10 Id−Vg simulation for acceptor traps with σvth = 7.8·10−18÷7.8·10−20 cm3/s.

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 725.11 Band diagram of a MIS structure with a trap level at energy Etr . . . . . . 735.12 C−V simulation for acceptor traps with energy levels at Etr = 0.3 eV+EV

and at Etr = EC − 0.3 eV. The curve with energy level in the middle ofthe band gap completely overlaps with the curve with Etr = EC − 0.3 eV.Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 75

5.13 Id−Vg simulation for acceptor traps with energy levels at Etr = 0.3 eV+EV

and at Etr = EC − 0.3 eV. The curve with energy level in the middle ofthe band gap completely overlaps with the curve with Etr = EC − 0.3 eV

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 755.14 C − V simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 765.15 Id − Vg simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.

Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 765.16 C − V simulation for acceptor traps, donor traps and no traps. Hysteresis

is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.17 Id−Vg simulation for acceptor traps, donor traps and no traps. Hysteresis

is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

vi LIST OF FIGURES

5.18 C − V simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 78

5.19 Id − Vg simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.Hysteresis is counter-clockwise. . . . . . . . . . . . . . . . . . . . . . . . . 78

*

List of Tables

3.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Electrical parameters of the devices. . . . . . . . . . . . . . . . . . . . . . . 40

4.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Extracted fitting parameter β. . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Fitting parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Geometry parameters for MIS and bottom contact OTFT. . . . . . . . . . 665.2 Material parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.3 Trap parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Gate sweep. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

*

vii

viii LIST OF TABLES

Abstract (Italian)

Il materiale maggiormente studiato ed impiegato nell’elettronica è indubbiamente il silicio.Le sue proprietà e i costi ad esso associati lo rendono un candidato ideale per la maggiorparte delle necessità dell’elettronica odierna. Tuttavia, diversi altri materiali sono statistudiati negli anni passati. Nel 1977, Alan J. Heeger, Alan G. MacDiarmid e HidekiShirakawa scoprirono un nuovo polimero su base carbonio altamente conduttivo: l’oxidizediodine-doped polyacetylene. Per la loro scoperta, che è stata una delle pietre miliari piùimportanti per l’elettronica organica, sono stati insigniti del Premio Nobel per la chimicanel 2000.

I semiconduttori organici hanno consentito a scienziati ed ingegneri di sviluppare dispo-sitivi con caratteristiche innovative e costi ridotti, rendendo questa tecnologia particolar-mente interessante per diversi settori dell’elettronica. I semiconduttori organici possonoessere realizzati e processati a temperatura ambiente, rendendo la loro produzione piùfacile e conveniente rispetto al silicio e agli altri semiconduttori inorganici; possono esseretrasparenti, flessibili e sviluppati su grandi aree o su geometrie non planari; possono essererealizzati anche dispositivi completamente plastici.

Le loro prestazioni, comparate con quelle del silicio, sono tuttavia inferiori: la mobilitàdei portatori è ordini di grandezza inferiore, anche se questa è notevolmente cresciutanegli ultimi anni; sono particolarmente sensibili alle condizioni ambientali, specialmenteall’atmosfera (ossigeno e umidità); le loro prestazioni decadono col tempo.

Nonostante questi inconvenienti, il grande potenziale dell’elettronica organica ha por-tato ad un’intensa attività di ricerca, sia teorica che sperimentale, le cui origini risalgonoalla fine degli anni Settanta, periodo in cui avvennero le prime scoperte. In questa tesidi dottorato viene sviluppato un quadro teorico che descriva l’elettronica dei transistororganici.

Nel capitolo 2 viene presentata una panoramica dell’argomento: a partire dalle pro-prietà fondamentali dei semiconduttori organici, il capitolo si sviluppa per fornire unaapprofondita analisi della fisica e dei modelli elettrici che descrivono il comportamentodei transistor organici a film sottile.

Nel capitolo 3 vengono modellati i transistor organici a film sottile con geometria

ix

x ABSTRACT (ITALIAN)

cilindrica. Questa particolare geometria ben si adatta a nuove applicazioni, come sistemitessili intelligenti per funzioni di monitoraggio biomedicale, interfacce uomo macchina e,più in generale, applicazioni e-textile. Viene presentato anche un confronto con risultatisperimentali.

Il capitolo 4 presenta un modello per transistor organici a film sottile a canale corto.Il riscalamento delle dimensioni nei dispositivi organici conduce a differenze significative,rispetto alle teorie introdotte nel capitolo 2, nelle caratteristiche del dispositivo, con effettiche possono già essere rilevanti per dispositivi con lunghezze di canale di dieci micron.Vengono considerati la mobilità dipendente dal campo ed effetti di carica spaziale, chelimita la corrente; viene inoltre fornito un confronto con i risultati sperimentali.

Il capitolo 5 analizza il comportamento dei transistor organici come funzione del tempo.Vengono esposti i modelli teorici che consentono di descrivere fenomeni di carica e scaricadi stati trappola, e vengono successivamente simulati per differenti parametri. Al variaredi tali parametri, vengono studiate le curve di trasferimento, per i transistor organici afilm sottile, e le curve capacità-tensione per le strutture metallo-isolante-semiconduttore.

Il capitolo 6 conclude questa tesi di dottorato. Vengono qui riassunti i risultati e leconsiderazioni più importanti, così come vengono esposti possibili future linee di ricercasull’argomento.

Chapter 1

Introduction

The most studied and widely employed material for electronics is undoubtedly silicon.Its semiconductor properties and its associated costs make it an ideal candidate for mostof the needs of today electronics. However, several different materials have been studiedin the last years. In 1977, Alan J. Heeger, Alan G. MacDiarmid, and Hideki Shirakawadiscovered a new, carbon based, highly-conductive polymer: the oxidized, iodine-dopedpolyacetylene. For their discovery, which was one of the most important milestones fororganic electronics, they were jointly awarded the chemistry Nobel Prize in 2000.

Carbon based semiconductors allowed scientists and engineers to develop devices withnovel features and reduced costs, making this technology become a premier candidate forseveral sectors of electronics. Organic semiconductors can be manufactured and processedat room temperature, making their production easier and cheaper than for conventionalsilicon and inorganic semiconductors; they can be transparent, flexible and developed overlarge areas or non-planar geometries; completely plastic devices can be realized.

Their performances, compared to silicon, are however inferior: carrier mobility is or-ders of magnitude lower, even if this largely increased in the last years; they are highlysensitive to environmental conditions, especially to the atmosphere (oxygen and humid-ity); their performances decrease over time.

Despite these drawbacks, the great potential of organic electronics has led to extensivetheoretical and experimental research since the first discoveries during the late seventies.In this doctoral thesis, a theoretical framework for the electronics of organic transistorsis developed.

In chapter 2 an overview of the topic is presented: starting from the fundamentalproperties of the organic semiconductors, the chapter develops to provide a thoroughanalysis of the underlying physics and the electrical models which describe the behaviorof organic thin film transistors.

In chapter 3, organic thin film transistors with a cylindrical geometry are modeled.

1

2 CHAPTER 1. INTRODUCTION

This particular geometry is well suited for novel applications, like smart textiles systemsfor biomedical monitoring functions or man-machine interfaces and, more in general, e-textile applications. A comparison with experimental results are presented.

Chapter 4 features a model for organic thin film transistors with short channel. Scalingof the dimensions in organic devices leads to significant deviations in the electrical charac-teristics of the devices from the theory exposed in chapter 2, with effects that can alreadybe relevant for devices with channel length of ten microns. Field-dependent mobility andspace charge limited current effects are considered, and a comparison with experimentalresults is also presented.

Chapter 5 analyzes the behavior of organic transistors as a function of time. The-oretical models that describe trap recharging are exposed and simulated for differentparameters sets. For each simulation, transfer curves for organic thin film transistors andcapacitance-voltage for metal-insulator-semiconductor structures are analyzed.

Chapter 6 concludes this doctoral thesis. The most important achievements and con-siderations are here summarized, and possible future activities on the topic are exposed.

Chapter 2

From charge transport to organictransistors

This chapter provides an introduction to the topic, starting from the charge transportin semiconductors up to the models which describe the behavior of planar organic thinfilm transistors. Section 2.1 describes how conjugated polymers can conduct a currentand the most important theories regarding charge transport; section 2.2 gives an overviewof the most important organic semiconductors whereas, from section 2.3 on, a detaileddescription of the transistor models is provided. The chapter ends with a short generaldescription of the drift diffusion model.

2.1 Charge transport in conjugated polymers

Materials employed as organic semiconductors are conjugated polymers, i.e. sp2-hybridizedlinear carbon chains. This kind of hybridization is the one responsible for giving (semi)-conducting properties to organic materials.

Starting from the electron configuration of a single carbon atom, which is 1s2 2s2 2p2,when multiple carbons bond together different molecular orbital can be originated, as theelectron wavefunctions mix together [Vol90] [Bru05].

While 1s orbitals of the carbons do not change when the atoms are bonded, 2s or-bitals mix their wavefunctions with two of the three 2p orbitals, leading to the electronconfiguration that is reported in Figure 2.1. Three sp2 orbitals are formed and lie on themolecular plane, at a 120 angle to each other, leaving one p orbital which is orthogonalto the molecular plane. Hybrid sp2 orbitals can then give origin to different bonds. sp2

orbitals from different carbon atoms which lie on the molecular plane form strong covalentbonds, which are called σ bond, whereas p orbitals which are orthogonal to the molecularplane can mix to give less strong covalent bonds, which are called π bonds. The π bonds

3

4 CHAPTER 2. FROM CHARGE TRANSPORT TO ORGANIC TRANSISTORS

plane of thesp orbitals2

p orbitalz

p orbitalz

- bond

- bond

- bond

Figure 2.1: sp2 hybridization of two carbon atoms. sp2 orbitals lie on the same plane andbond into a σ-bond, pz orbitals are orthogonal to the plane and bond into a π-bond.

constitute a delocalized electron density above and below the molecular plane and are re-sponsible for the conductivity of the molecule, as the charge carriers move through thesebonds.

Depending on the sign of the wavefunction of the orbitals, these can be π (bonding)or π∗ (anti-bonding) orbitals, the latter having a higher energy level, as shown in fig.2.2. When multiple molecules are considered, the energy levels of the orbitals give originto energy bands, in analogy to what happens in inorganic semiconductors: the edge ofthe valence band corresponds then to the Highest Occupied Molecular Orbital (HOMO),whereas the edge of the conduction band corresponds to the Lowest Unoccupied MolecularOrbital (LUMO). The difference between the energy of the HOMO and the energy of theLUMO is the energy gap Eg of the organic material and usually 1.5 < Eg < 4 eV [ZS01].

Charge transport in organic semiconductor is, however, quite different with respectto silicon and other mono-crystalline inorganic semiconductors. The periodic lattice ofthese materials and their very low density of defects allows one to accurately describe thecharge transport by means of delocalized energy bands separated by an energy gap. Mostof the organic semiconductors, on the other hand, are amorphous and rich in structuraland chemical defects, therefore requiring conventional models for charge transport tobe adapted and extended; moreover, charges can move, with different mobilities, withinthe molecular chain (intra-chain), between adjacent molecules (inter-chain), or betweendifferent domains, generally referred as grains (inter grain).

2.1. CHARGE TRANSPORT IN CONJUGATED POLYMERS 5

+-

+

-

+

-

+

-

p orbital p orbital

Bonding molecular orbital

Anti-Bonding molecular orbital

Energ

y

Figure 2.2: Bonding of pz orbitals: depending on the sign of the wavefunctions, π orbitalswith lower energy or π∗ orbitals with higher energy can be originated.

2.1.1 Hopping between localized states

The presence of defects and the non-crystalline structure of the organic polymers leadsto the formation of localized states. In order to move, charges must hop between theselocalized states and overcome the energy difference between them, emitting or adsorbingphonons during intra-chain or inter-chain transitions. Attempts to model hopping ininorganic semiconductors are reported in [Mot56] [Con56], later followed by Miller andAbrahams [MA60], who described the rate of single phonon jumps.

In 1998 Vissenberg and Matters [VM98] developed a theory for determining the mo-bility of the carriers in transistors with amorphous organic semiconductors. They pointedout that the transport of carriers is strongly dependent on the hopping distances as wellas the energy distribution of the states. At low bias, the system is described as a resistornetwork, assigning a conductance Gij = G0 exp (−sij) between the the hopping site i andthe site j, where G0 is a prefactor for the conductivity and

sij = 2αRij +|Ei − EF|+ |Ej − EF|+ |Ei − Ej|

2kT. (2.1)

The first term on the right-hand side describes the tunneling process, which depends onthe overlap of the electronic wave functions of the sites i and j, EF is the Fermi energy


and Ei and Ej the energies of the sites i and j. a In a lowest-order approximation,this tunneling process may be characterized by the distance Rij between the sites andan effective overlap parameter α. The second term takes into account the activationenergy for a hop upwards in energy and the occupational probabilities of the sites i and j.Starting from this expression, with the percolation theory, they can relate the microscopicproperties of the organic semiconductors to the effective mobility of the carriers in atransistor. More details are provided in section 2.4.3, where mobility models for theorganic transistors are discussed.

2.1.2 Multiple trapping and release model

The Multiple Trapping and Release (MTR) model has been developed by Shur and Hack[SH84] to describe the mobility in hydrogenated amorphous silicon. Later, Horowitz etal. extended it to organic semiconductors [HHD95] [HHH00].

The model assumes that charge transport occurs in extended states, but that most ofthe carriers injected in the semiconductor are trapped in states localized in the forbiddengap. These traps can be deep, if their energy level is near the middle of the band gap, orshallow, if they are located near the conduction or valence band. This is exemplified infig. 2.3.

HOMO

LUMO

shallow

deep

E

DOS

few kT

E /2g

Eg

Figure 2.3: Distribution of trap states in the band gap. Shallow traps energy is just fewkT over the valence band, for a p-type semiconductor (conduction band for n-type).

The model shows a dependence of the mobility of the carriers on temperature, theenergetic level of the traps, as well as on the carrier density (and therefore on the applied

2.2. MATERIALS 7

voltage to a device). For a single trap level with energy Etr the drift mobility is

µD = µ0α exp

(−Etr

kT

). (2.2)

The temperature dependance decreases with temperature itself and, for low values, thesize of the grains of the semiconductor must be accounted. The polycrystalline semicon-ductor is described as trap-free grains separated by boundaries with high trap density. Iftheir size is lower than the Debye length, the distribution of the traps can be considereduniform. However, if the grains are much larger than the Debye length, charges movethrough the grain boundaries. At high temperature, this occurs via thermionic emissionand a dependence on the temperature is found; at low temperatures, the charges cantunnel through the grain boundaries, so the mobility becomes temperature independent.At intermediate temperatures, charge transport is determined by thermally activated tun-neling.

2.1.3 The polaron model

The polaron model was introduced by Yamashita et al. [YK58] in 1958 for inorganicsemiconductors. Later, it was extended by Holstein [Hol59] and by Fesser et al. [FBC83]to molecular crystals and conjugated polymers.

Charge transport in organic semiconductors can be described by means of polarons:a polaron is a quasiparticle composed of an electron plus its accompanying polarizationfield. In organic polymers, they result from the deformation of the conjugated chain underthe action of the charge. Holstein [Hol59] proposed a model to determine the mobilityµ of polarons in the semiconductor, as a function of the lattice constant a, the electrontransfer energy J , the reduced mass of the molecular site M , the frequency of the harmonicoscillators associated to the molecules ω0, the polaron binding energy Eb = A2/ (2Mω2

0)

and the temperature T :

µ =

√π

2

qa2

~J2

√Eb

(kT )−3/2 exp

(− Eb

2kT

). (2.3)

Equation (2.3) holds for temperatures T > Θ, where Θ is the Debye temperature definedas kΘ = ~ω0.

2.2 Materials

Semiconductors are generally referred as n-type or p-type if their carriers are electrons orholes, respectively. For organic semiconductors, the same definitions hold, although the


most widely used and studied semiconductors are p-type. This is mainly because of theirhigher stability in air and higher mobility (with respect to n-type semiconductors) whenthey are employed for organic transistors. On the contrary, n-type semiconductors arehighly sensitive to oxigen and water, for the presence of carbanions in their structure.

Figure 2.4: p-type organic semiconductors.

2.2. MATERIALS 9

Figures, 2.4, 2.5 and 2.6 [ZS07] show the most common organic semiconductors. Fig-ure 2.4 shows p-type semiconductors. Pentacene is probably the most used, as it offersthe best mobility among all organic semiconductors (hole mobility in OFETs of up to5.5 cm2/Vs have been reported). It is a polycyclic aromatic hydrocarbon consisting of5 linearly-fused benzene rings. Also shown in fig. 2.4 is P3HT, which results from thepolymerization of thiophenes, a sulfur heterocycle. Just like pentacene, also P3HT hasbeen of interest because of its high carrier mobility, mechanical strength, thermal sta-bility and compatibility with fabrication process. Figure 2.5 shows semiconductors withpredominantly n-channel behavior, in transistors with SiO2 as a gate dielectric and goldsource-drain electrodes. Figure 2.6 shows semiconductors with ambipolar behavior (theycan have both p- and n-type conduction).


Figure 2.5: n-type organic semiconductors.

2.2. MATERIALS 11

Figure 2.6: Ambipolar organic semiconductors: (a) oligothiophene/ fullerene dyad andoligothiophene/fullerene triad; (b) 9-(1,3- dithiol-2-ylidene)thioxanthene-C60 system (n =

6); (c) poly(3,9-di-tert-butylindeno[1,2-b]fluorene) (PIF); (d) the near-infrared absorbingdye bis[4-dimethylaminodithiobenzyl]nickel (nickel di- thiolene); (e) quinoidal terthio-phene (DCMT).


2.3 Organic field effect transistors

Organic transistors are three terminal devices in which the current flow going betweensource and drain is modulated by a gate potential. A major difference with commonlyused inorganic transistors is that no inversion layer is formed, but the conduction oc-curs by means of the majority carriers, which accumulate at the semiconductor/insulatorinterface.

Possible transistor structures are shown in fig 2.7 and 2.8. Both have a substratewhich acts as mechanical support for the structure. Over it, there is an insulating di-electric film, under the which a gate contact is realized. Top contact structures have thesemiconducting layer all over the insulator, with the source and drain contacts lying ontop of the semiconductor; on the contrary, bottom contact devices have their contactsunder the semiconductor layer. Moreover, both structures can also be realized withoutthe substrate, if the mechanical support to the structure is provided by the insulating filmitself (free standing devices).

Source

Gate

Substrate

Insulator

Semiconductor

Drain

Figure 2.7: Top contact transistor structure.

Source

Gate

Substrate

Insulator

SemiconductorDrain

Figure 2.8: Bottom contact transistor structure.

2.3.1 Analytical derivation of the current

The determination, by analytical means, of the drain current for any applied voltages, hasbeen object of extensive research over the years. For example, Horowitz et al. [HHB+98],assuming a constant mobility, developed a model which can describe the behavior of the

2.3. ORGANIC FIELD EFFECT TRANSISTORS 13

transistor in the linear and saturation region. Colalongo et al. [CRV04] and Li et al.[LK05] included also the effect of the variable range hopping mobility. The same effecthas been taken into account by Calvetti et al. [CSKVC05] to model the current in thesubthreshold region.

Other contributions to the modeling of the organic transistors, including the effects oftraps and ambipolar devices, come also from Stallinga et al. in [SG06a, SG06b].

In this section, we will describe a model based on constant mobility and doping, inlinear and saturation region. It is mostly based on [HHB+98]. Extensions to the modelwill be provided in the other sections of this thesis.

Device characteristics

Figure 2.9 shows the basilar structure for the transistor to be modeled. It is a free-standingbottom contact structure.

ds

di

L

Source

Gate

Insulator

Semiconductor Draindx

0

y

Figure 2.9: Geometry of the bottom contact transistor.

The length of the channel is L, its width is Z. The dielectric layer has thickness di

and dielectric permittivity εi, with an associated capacity Ci = εi/di whereas the semi-conductor film has thickness ds and dielectric permittivity εs, with an associated capacityCs = εs/ds. We will assume an n-type semiconductor with doping N and a density of freecarriers n0 ' N ; an analogous model can be derived for p-type semiconductors changingthe sign of the current and the applied voltages.

The threshold voltage of the device, which accounts for different workfunctions betweenthe semiconductor and the gate and for possible charges located in the insulator or in theinsulator/semiconductor interface, is set to be zero, to simplify notation. Since it justdetermines a shift of the gate voltages, non-zero threshold voltages can be included in themodel substituting Vg with Vg − Vth.


Linear region

The total drain current Id in the linear region (Vd < Vg) can be expressed as the sum ofthe bulk current originating from the free carriers in the semiconductor and the currentdetermined by the carriers in the accumulation layer. Therefore

Id

Zµ= − [Qg(x) + Q0]

dV

dx, (2.4)

where Q0 is the surface density of the free carriers given by

Q0 = ±qn0ds, (2.5)

wherein the sign of the right hand side of (4.3) is that of charge of the carriers (plus forholes, minus for electrons). The surface density of the carriers in the accumulation layerQg(x) which forms as a capacitive effect, is

Qg(x) = −Ci [Vg − Vs(x)− V (x)] , (2.6)

wherein Vs(x) is the ohmic drop in the bulk of the semiconductor, which can be generallyneglected [HHK98], and V (x) is the potential with respect to the source at the coordinatex. Then we can state that

Id

Zµ= Ci [Vg + V0 − V (x)]

dV

dx, (2.7)

where V0 = −Q0/Ci. Equation (4.5) can be integrated from V (x = 0) = 0 V to V (x =

L) = Vd to obtain

Id

∫ L

0

dx = IdL = Z

∫ Vd

0

µCi (Vg − V + V0) dV, (2.8)

which leads, for a constant mobility, to

Id =Z

LµCi

[(Vg + V0) Vd −

V 2d

2

]. (2.9)

The transconductance in the linear zone, from which the field effect mobility can beextracted, can then be defined as

gm =

(∂Id

∂Vg

)Vd=const

=Z

LµCiVd. (2.10)

2.3. ORGANIC FIELD EFFECT TRANSISTORS 15

Saturation region

When the drain voltage exceeds the effective gate voltage Vd > Vg the accumulationlayer near the drain changes to a depletion layer. The depletion layer will extend from acoordinate x0 such that V (x0) = Vg to the drain contact (x = L). At those coordinateswhere there is the depletion layer, the only contribution to the current is given by the freecarriers, which are only in the volume of the semiconductor which has not been depleted.Then

Iddx = Zqµn0 [ds −W (x)] dV, (2.11)

wherein W (x) is the width of the depletion region, which can be determined solving thePoisson equation [HHK98]

d2V

dy2= −qN

εs

, (2.12)

subject to the boundary conditions

V (W ) = 0, (2.13)dV

dy

∣∣∣∣y=W

= 0. (2.14)

The solution isV (x) =

qN

2εs

(x−W )2 . (2.15)

Therefore, at the semiconductor/insulator interface the potential is

Vs =qN

2εs

W 2. (2.16)

The voltage drop at the insulator is

Vi =qNW

Ci

, (2.17)

so the following Kirchhoff equation holds:

Vg + V (x) = Vi + Vs. (2.18)

The width of the depletion region can then be determined as

W (x) =εs

Ci

√

1 +2C2

i [V (x)− Vg]

qNεs

− 1

. (2.19)

Since we have assumed that the accumulation layer extends up to a point where V (x0) =

Vg and since beyond x0 only the non-depleted free carriers add to the current, we canwrite the latter as the sum of these two contributions

Id =Z

LµCi

∫ Vg

0

(Vg + V0 − V ) dV +Z

Lµqn0

∫ Vdsat

Vg

(ds −W ) dV. (2.20)


The second integral on the right end side means that we account for all the free carrierswhich are in the semiconductor out of the depletion region, whose extension is W (Vg) = 0

at the end of the accumulation layer and is W (Vdsat) = ds when the device reaches thesaturation voltage. Changing the integration variable in (2.20) from V to W leads to

Id =Z

LµCi

∫ Vg

0

(Vg + V0 − V ) dV +Z

Lµ

q2n0N

εs

∫ ds

0

(ds −W )

(W +

εs

Ci

)dW =

=Z

Lµ

[Ci

(V 2

g

2+ V0Vg

)+

q2n0N

εs

d3s

6

(1 +

3Cs

Ci

)], (2.21)

The pinch-off voltage Vp, i.e. the gate voltage for which the depletion region extension isequal to ds, can be obtained by substituting Vp = Vg − V (x) in (2.19). This leads to

Vp = ±qNd2s

2εs

(1 + 2

Cs

Ci

)' qNds

Ci

, (2.22)

wherein in the approximated expression the flat band voltage has been neglected andds di is reasonably assumed. If also n0 = N is assumed, then Vp = V0 and

Idsat =Z

2LµCi (Vg − V0)

2 . (2.23)

2.4 Mobility models

As different theories for charge transport in organic semiconductors have been developed,several models for the mobility in organic field effect transistors exist. Most of themdetermine a mobility which depends on the temperature and the electric field in thesemiconductor. The most common are hereafter reported.

2.4.1 Multiple trap and release mobility

To determine the MTR mobility, the width of the accumulation layer is first estimatedsolving the Poisson equation

d2V

dx2=

qn0

εs

expqV (x)

kT, (2.24)

where n0 is the density of carrier at the equilibrium. For a semi-infinite semiconductor,the carrier density n(x) is found to be

n(x) ' 2εskT

q2 (x + La)2 , (2.25)

2.4. MOBILITY MODELS 17

where La is the effective length of the accumulation layer. Since the total induced chargedis almost equal to the accumulation charge, then∫ ∞

0

qn(x) dx ' CiVg (2.26)

, and the effective length is

La =2εskT

qCiVg

. (2.27)

This length is usually around a nanometer, so the accumulation layer lies within the firstatomic monolayer.

The MTR model assumes that the charge σ, induced by the gate voltage Vg, splits ina free charge σf and a trapped charge σt, the latter being much greater than the formerσt σf . Therefore

σt ' σ ' CiVg. (2.28)

Boltzmann statistics says that the free charge is given by

σf = σf0 expqVs

kT, (2.29)

wherein σf0 = qNC exp [− (EC − EF) /kT ] is the free charge at the equilibrium. NC is thesurface density of charge at the conduction band (for n-type semiconductor, valence bandfor p-type), whereas EF is the Fermi level at the equilibrium. the measured mobility inthe device is

µ = µ0σf

σ. (2.30)

Therefore

EC − EqF = EC − EF − qVs = kT lnqµ0NC

µCiVg

, (2.31)

where EqF is the quasi-Fermi level. The trapped charge σt depends on the density ofstates so that

σt =

∫ +∞

−∞Ntr(E)f(E) dE (2.32)

and if the Fermi distribution function f(E) is varying slow, it can be approximated witha step function so that

Ntr(E) =dσt

dE. (2.33)

Assuming an exponential distribution of states, like

Ntr(E) =Nt0

kTc

exp

(−EC − E

kTc

), (2.34)


wherein Nt0 is the density of traps at the equilibrium and Tc is a characteristic temperatureto account for the steepness of the exponential distribution. It follows that the trappedcharge is

σt = σt0 expqVs

kTc

= σt0X, (2.35)

wherein σt0 = Nt0 exp [− (−EC − EF) /kTc] is the trapped charge at the equilibrium (Vg =

0) and X = exp (qVs/kTc). The free charge can be now written as

σf = σf0X`, (2.36)

where ` = Tc/T . Combining the previous equations leads to

µFET = µ0NC

Nt0

(CiVg

qNt0

)`−1

. (2.37)

When a threshold voltage Vth is included, the mobility can be written as

µFET = µ0 (Vg − Vth)α , (2.38)

where µ0 and α are parameters usually extracted by means of fitting of the experimentaldata.

The MTR model does not always return correct results, as it does not include theeffects of the grain nature of many organic materials, and it fails when the devices operateat low temperatures.

2.4.2 Grain Boundaries

Horowitz accounted for the grain nature of the semiconductor in the second part of hispaper [HHH00]. His analysis assumes that the mobility is limited by the grain boundaries,in which the conductivity is much lower than in the crystal grain. This phenomenon hasbeen already studied in the past for inorganic polycrystalline materials [OP80], whosemobility is found to increase with the size of the grains.

Being the grains and the boundaries connected in series, the mobility can be expressedas

1

µ=

1

µb

+1

µg

, (2.39)

where µb is the mobility of the boundary and µg is the mobility of the grain, whosesize is supposed to be much larger than the boundary region. Between the grains, aback-to-back Schottky barrier is formed. Depending on the temperature, the current willbe modeled differently, as thermionic emission dominates at high temperatures, whereastunnel transport can dominate at lower ones.


At high temperatures, the current density through a boundary is given by [Hee68]

jb =1

4qngv exp

(−Eb

kT

)[exp

(qVb

kT

)− 1

], (2.40)

where ng is the carrier concentration in the grains, v =√

8kT/πm∗ is the electron thermalvelocity, Eb is the barrier height and Vb is the voltage drop across a grain boundary. Forgrain boundaries of the same size, the source-drain voltage Vd is equally divided betweenthe barriers and Vb = Vd(l/L), where l is the length of the grain and L is the distancebetween source and drain. Equation (2.40) can be expanded to the first order for high T

so thatjb ' qngµb

Vd

L, (2.41)

wherein the mobility in grain boundary is

µb = µ0 exp

(−Eb

kT

)(2.42)

andµ0 =

qvl

8kT, (2.43)

where the factor 1/2 that appears between (2.40) and (2.43) is related to the two Schottkybarriers at the sides of the grain boundary. It should be noted that the model can onlyqualitatively fit the experimental data [HHH00].

At low temperatures, the mobility can be modeled starting from tunnel transport ina metal-semiconductor junction [PS66]:

jb = j0 exp

(qVb

E00

), (2.44)

where

j0 = j00(T ) exp

(− Eb

E00

)(2.45)

and

E00 =~q

2

√N

m∗εs

. (2.46)

Equation (2.44) holds for kT < E00 and can be expanded to the first order so that themobility at low temperature becomes

µ = µ00(T ) exp

(− Eb

E00

), (2.47)

where µ00(T ) varies slowly with the temperature. For low temperatures, the mobility µ

is no longer thermally-activated.


2.4.3 Variable range hopping

VRH mobility has already been introduced in section 2.1.2 and refers to [VM98]. Herefurther details are reported to obtain a power law for the mobility in organic transistors.

The model can be developed starting from the conductivity given by the percolationtheory [AHL71] for a semiconductor with an exponential distribution of states. Theconductivity is

σ(δ, T ) = σ0

[πNtδ(T0/T )3

(2α)3BcΓ(1− T/T0)Γ(1 + T/T0)

]T0/T

(2.48)

where Nt is the number of states per unit volume, T0 is a parameter that accounts forthe width of the exponential distribution of states, δ is the carrier occupation at thetemperature T , α is the effective overlap parameter, Bc is a parameter given by thepercolation theory and Γ is the Γ-function. The carrier occupation δ is given by theBoltzmann distribution and is related to the gate-induced applied voltage by

δ(x) = δ0 exp

[qV (x)

kT

](2.49)

where δ0 is the value of the distribution far from the semiconductor/insulator interface,where V (x) = 0. The relation between the potential and the δ(x) is determined by thePoisson equation. For an accumulation layer δ(x) δ0, the electric field is [HHD95]

E2(x) = 2kT0Ntδ(x)/εs. (2.50)

Gauss law gives the electric field at the interface as

E(0) ' CiVg/εs. (2.51)

The current in the linear region for a transistor with source-drain potential Vd, semicon-ductor thickness t, width Z and length L is

I =WVd

L

∫ t

0

σ [δ(x), T ] dx. (2.52)

The field-effect mobility can be determined from the conductance as

µFET =L

CiWVd

∂I

∂Vg

=

=σ0

q

[π(T0/T )3

(2α)3BcΓ(1− T/T0)Γ(1 + T/T0)

]T0/T[

(CiVg)2

2kTεs

]T0/T−1

, (2.53)

where it has been assumed that the semiconductor layer in large enough so that V (t) = 0.Again, including also the threshold voltage Vth, the effective mobility can be expressed as

µFET = µ0 (Vg − Vth)α . (2.54)


A similar approach [LK05] allows the determination of the current in the triode ansaturation region. Given the following coefficients,

β = σ0

√2εskT

δ0Nt

kT

q (T − 2T0), (2.55)

γ =(2α)3 Bc2kTεs

C2i (T0/T )3 sin (πT/T0)

, (2.56)

the current in the triode region is

Id = βW

L

[(Vg − Vfb

γ

)2T0/T

−(

Vg − Vfb − Vd

γ

)2T0/T]

(2.57)

and in the saturation region is

Id = βW

L

(Vg − Vfb

γ

)2T0/T

. (2.58)

2.4.4 Poole-Frenkel mobility

Poole-Frenkel mobility was first theorized in 1938 by Frenkel [Fre38] to explain the increaseof conductivity in insulators and semiconductor when high fields are applied. Its form is

µ = µ0 exp(γ√E)

, (2.59)

where µ0 and γ are parameters which depend on the physics of the considered systemand E is the electric field. Poole-Frenkel mobility has been extensively studied for model-ing charge transport in organic LEDs with disordered semiconductors [DPK96] [SPG89][NDK+98], but it can also describe several experimental results for organic field effecttransistors which show a variation of the mobility with the source-drain electric field[WFBD07] [HRG+07] [HN07].

In disordered organic semiconductors charge transport occurs mainly by hopping be-tween nearby localized states which are induced by disorder. In a transistor, the physicaleffect of the source-drain electrical field is then to effectively reduce the hopping barrier.Assuming also a Coulomb potential type for hopping barrier, the hopping probability andthe mobility, will have a dependence on electrical field which follows a Poole-Frenkel lawlike (2.59) with zero-field mobility µ0 given by

µ0 = µi exp

(− ∆

kT

), (2.60)

where µi is the intrinsic mobility at zero hopping barrier and ∆ is the zero-field hoppingbarrier, also known as low field activation energy. Equation (2.59) can be rewritten as

µ = µi exp

(β√E −∆

kT

). (2.61)


Equation (2.61) has been modified by Gill [Gil72] to increase agreement with experimentaldata substituting T with an effective temperature Teff so that

1

Teff

=1

T− 1

T0

, (2.62)

where T0 is a fitting parameter.

2.5 Drift diffusion model

All the models proposed in this chapter are based on the drift-diffusion model. Even ifit was originally developed for inorganic semiconductors, it has been extended to accountfor the organic semiconductors characteristics. In this section we report a more generaldescription of the drift-diffusion model, providing the differential equations for the poten-tial and the current, as well as the boundary conditions which are generally applied byconventional drift-diffusion solvers available, like Sentaurus by Synopsys [sen] or Atlas bySilvaco [atl] . The drift diffusion model is based on the Poisson equation:

∇ · (ε∇V ) = q (n− p + NA −ND + ntr) , (2.63)

where V is the potential in the structure, ε is the electric permittivity of the material, q

is the elemental charge, n and p are the density of holes and electrons, respectively, NA

and ND are the concentrations of ionized acceptors and donors, respectively, and ntr isthe density of occupied traps.

The continuity equations are

∇ · Jn = +qRn + q∂n

∂t, (2.64)

∇ · Jp = −qRp − q∂p

∂t, (2.65)

where Jn and Jp are the current densities and Rn and Rp are the net generation-recombinationrate.

In the DD model the currents of electrons and holes are described as the sum of twocontributions, namely a drift component, proportional to the electrostatic field E = −∇V

and a diffusion component, proportional to the gradient of the carrier density:

Jn = −qnµn∇V + qDn∇n, (2.66)

Jp = −qpµp∇V − qDp∇p. (2.67)

The density of carriers is described by means of Boltzmann statistics:

2.5. DRIFT DIFFUSION MODEL 23

n = NC exp

(EqF,n − EC

kT

), (2.68)

p = NV exp

(EV − EqF,p

kT

), (2.69)

where NC and NV are the effective density of states, EqF,n = −qΦn and EqF,p = −qΦp arethe quasi-Fermi energies for electrons and holes, Φn and Φp are the quasi-Fermi potentials,respectively. EC and EV are the conduction and valence band edges, defined as:

EC = −χ− q (V − φref) , (2.70)

EV = −χ− Eg − q (V − φref) , (2.71)

where χ is the electron affinity and Eg the band gap. The reference potential φref can beset equal to the Fermi potential of an intrinsic semiconductor. Then (2.68) and (2.69)become

n = ni expq (V − φn)

kT, (2.72)

p = ni expq (φp − V )

kT, (2.73)

where ni =√

NCNV exp (−Eg/2kT ) is the intrinsic density.

2.5.1 Boundary conditions

Ohmic contacts

For ohmic contacts charge neutrality and equilibrium are assumed:

n0 − p0 = ND −NA, (2.74)

n0p0 = n2i . (2.75)

Applying Boltzmann statistics we obtain

V = φF +kT

qasinh

(ND −NA

2ni

), (2.76)

n0 =

√(ND −NA)2

4+ n2

i +ND −NA

2, (2.77)

p0 =

√(ND −NA)2

4+ n2

i −ND −NA

2, (2.78)

where n0 and p0 are the electron and hole equilibrium densities and φF is the Fermipotential at the contact, that is equal to the applied potential.


Schottky contacts

For Shottky contacts, the following boundary conditions hold:

V = φF +kT

qln

(NC

ni

), (2.79)

Jn · n = vn

(n− nB

0

), (2.80)

Jp · n = vp

(p− pB

0

), (2.81)

nB0 = NC exp

(−qΦB

kT

), (2.82)

pB0 = NV exp

(−Eg + qΦB

kT

), (2.83)

where φF is the Fermi potential at the contact, that is equal to the applied potential,ΦB = φm−χ is the barrier height (the difference between the metal work-function φm andthe electron affinity χ), vn and vp are the thermionic emission velocities and nB

0 and pB0 are

the equilibrium carrier densities. Fig. 2.10 shows a band diagram for a Schottky contactbetween an n-type semiconductor and metal, including also the difference between themetal and the semiconductor workfunction φms.

EC

Vacuum level

EFEF

qs

qm

qms

q

qB

EV

Figure 2.10: Band diagram for a Schottky contact between an n-type semiconductor andmetal at the thermal equilibrium.

Gate contacts

For gate contacts, the boundary condition for the potential is

V = φF − φms, (2.84)

2.5. DRIFT DIFFUSION MODEL 25

where φF is the Fermi potential at the contact and φms = φm−φs is the difference betweenthe metal and the semiconductor work-functions.


Chapter 3

Cylindrical thin film transistors

The evolution of modern device electronics has led to the realization of devices with severalgeometries, in order to fulfill different requirements. For instance, cylindrical geometriesare often used to obtain device size reduction without the occurrence of short channeleffects like in surrounding gate devices [AP97] [KHG01]; recently, cylindrical geometrieshave been used for producing thin film transistors by means of organic semiconductors [LS][MOC+06] with the aim of obtaining distributed transistors on a long yarn-like structuresuitable to be employed for e-textile applications. This field has recently attracted astrong interest for several novel applications that are potentially feasible thanks to thisnew technology: smart textiles systems for biomedical monitoring functions, man-machineinterfaces and more could be, in the near future, realized using innovative electron devicesand materials in a textile form.

Different approaches have led to textile transistors [MOC+06] [BRK+05], whose func-tionality is given by the particular topology and materials of the yarns used, or to weavepatterned transistors [LS]. In [MOC+06], an organic field effect transistor with a cylin-drical geometry was developed with mechanical features and size fully compatible withthe usual textile processes.

Each of the above mentioned devices was developed using organic materials but, inprinciple, these devices could also be produced with other semiconductors as, for instance,amorphous silicon.

To describe the electronic behavior of cylindrical thin film transistors, we have de-veloped a model that can be applied to whatever kind of semiconductor, focusing inparticular on the geometrical constraints that define the cylindrical geometry.

As any other organic thin film transistor (OTFT), the proposed devices operate in ac-cumulation mode. In the following the usual models known for planar OTFTs [HHB+98][HHK98] will be adapted to the new cylindrical geometry. Section 3.1 describes the struc-ture of the device; section 3.2 and 3.3 are dedicated, respectively, to the description of the

27

28 CHAPTER 3. CYLINDRICAL THIN FILM TRANSISTORS

model in the linear and in the saturation regions. Finally, in section 3.4, experimental re-sults for an organic semiconductor based device are shown and compared to the developedmodel.

3.1 Device structure

A schematic view of the device is shown in fig. 4.1. A metal cylinder, with radius rg,acts as the gate electrode, and is surrounded by an insulating layer, with thickness di andouter radius ri. The semiconductor surrounds the insulator with thickness ds and outerradius rs. The source and drain electrodes are the two external rings and their distance,i.e. the length of the conducting channel, is L.

rg

L

ri

rs

z

rg

ri

rs

Figure 3.1: Geometry of the cylindrical TFT.

In the following we aim at showing the effects of the geometry on the device character-istics. All non geometrical parameters, used in the proposed simulations, are fixed to thevalues reported in table 3.1. These values are plausible for OTFTs. However, the modelis general and is valid for different values of these parameters.

3.2. LINEAR REGION 29

Parameter Symbol Value

Dielectric permittivity εs 3ε0

Insulator permittivity εi 3ε0

Density of free carriers n0 1017 cm−3

Semiconductor doping N 1017 cm−3

Table 3.1: Simulation parameters.

3.2 Linear region

In this section the current-voltage equations of the devices, for the linear operation regime,will be determined. The main difference with a planar TFT is in the threshold voltage,which depends on the gate radius rg as it is related to the number of free carriers available.

Let D be the domain determined by the annulus of inner radius ri and outer radius rs

at any cross-section of the cylinder. Its area A is equal to

A = π(r2s − r2

i

).

The conductivity σ, averaged over the annulus surface, is given by (3.1)

σ =qµ

A

∫∫D

n(r)r dr dθ, (3.1)

where q is the elemental charge, µ is the carrier mobility and n(r) is the carrier densityat radius r. The elemental resistance dR of an elemental segment dz is given by

dR =1

σ

dz

A. (3.2)

For a TFT operating in the accumulation regime, in addition to the free carriers of thesemiconductor, the density of which is uniform and equal to n0, there are the carriersdetermined by the accumulation layer, localized at the interface between the insulatorand the semiconductor, at radius ri, with density na(r). Therefore, the total density ofcarriers is given by

n(r) = n0 + na(r). (3.3)

The density na(r) can be determined considering the capacitor with inner radius rg andouter radius ri, which has a capacitance per area unit

Ci =εi

ri ln(

ri

rg

) , (3.4)

where εi is the dielectric permittivity of the insulator. The voltage Vc applied to thecapacitor is given by

Vc = Vg − Vfb − V (z), (3.5)


where Vg is the gate voltage, Vfb is the flat band potential and V (z) is the potential atthe point z. Under the gradual channel approximation (i.e. the electric field along z isnegligible with respect to that along r), V (z) increases from 0 at the source to Vd at thedrain.

The density na is then given by

na(r) =CiVc

qδ (r − ri) , (3.6)

where a Dirac distribution is used in order to account for the superficial distribution ofthe carriers accumulated at the interface between the insulator and the semiconductor.

The substitution of (3.6) in (3.3) and in (3.1) leads to

σ =qµ

A

∫∫D

[n0 +

CiVc

qδ (r − ri)

]r dr dθ =

qµ

A

(An0 +

ZCiVc

q

), (3.7)

where Z is the width of the channel and is equal to

Z = 2πri. (3.8)

Thus, the elemental resistance dR is given by

dR =dz

qµ(An0 + ZCiVc

q

) . (3.9)

The drain current can Id can be determined starting from

dV = IddR =Iddz

qµ(An0 + ZCiVc

q

) . (3.10)

Integrating (3.10) between 0 and Vd with respect to V and between 0 and L with respectto z to obtain

Id =Z

LµCi

[qn0A

ZCi

Vd + (Vg − Vfb) Vd −V 2

d

2

]. (3.11)

Since a non-zero drain current flows in the device even if no bias is applied to the gate, athreshold voltage Vth can be introduced:

Vth = ±qn0A

ZCi

+ Vfb = ±qn0 (r2

s − r2i ) ln

(ri

rg

)2εi

+ Vfb. (3.12)

The sign of the right hand side accounts for the type of majority carriers involved (holesor electrons). The drain current becomes

Id =Z

LµCi

[(Vg − Vth) Vd −

V 2d

2

]=

2π

L

εi

ln(

ri

rg

)µ

[(Vg − Vth) Vd −

V 2d

2

]. (3.13)

3.2. LINEAR REGION 31

0 20 40 60 80 1001

1.05

1.1

1.15

1.2

1.25

Gate radius (μm)

Nor

mal

ized

thre

shol

d vo

ltage

di = 500 nm

di = 100 nm

Figure 3.2: Normalized threshold voltage vs. gate radius for Vfb = 0 V, and ds = 50 nm anddifferent insulator thicknesses. The curves are normalized with respect to the thresholdvoltages of planar devices with the same thicknesses.

Under the hypotheses that rg ds and rg di, which usually hold for any organicelectron device, (3.12) can be simplified to obtain

Vthp =qn0dsdi

εi

+ Vfb, (3.14)

wherein Vthp is also the expression for the threshold voltage for a planar TFT as describedby Horowitz in [HHB+98]. Fig. 3.2 shows the threshold voltages, for a cylindrical TFT asa function of the gate radius for different insulator thicknesses. The curves are normalizedwith respect to the threshold voltages of planar devices with the same thicknesses. It canbe seen that the voltages of the cylindrical devices asymptotically tend to the ones of theplanar TFTs (provided that the thickness of the insulator and the semiconductor are thesame in both cases). Moreover, it can be noticed that increasing the insulator thicknessalso increases the difference of the threshold voltage shift between a cylindrical and aplanar device.

One last remark is about contact resistance, which can reach values of MΩ [KSR+03],especially with organic polycristalline semiconductors. In a cylindrical TFT it is propor-tional to the length of the circumference with radius rs, and can be included in the model


of (3.13) by substituting Vd with Vd − 2RsId and Vg with Vg − RsId, where RsId is theohmic drop due to one contact resistance Rs:

Id =2πεiµ

L ln(

ri

rg

) [(Vg −RsId)− Vth] (Vd − 2RsId)−(Vd − 2RsId)

2

2

. (3.15)

A planar device can have any contact width, so its contact resistance can be small; ina cylindrical device this could be achieved only increasing rs, which cannot be performedwithout affecting also the other parameters of the device.

3.3 Saturation region

When the TFT enters the saturation region, the channel gradually depletes, so that nofree carriers are available in the depletion region and the accumulation layer is limitedto the part of the interface between insulator and semiconductor which has not beendepleted. Let rd be the radius of the extremity of depletion region and W be its width,so that

rd = ri + W. (3.16)

To determine the extension of the depletion region, so that one can find the voltage drop ϕs

across the semiconductor, the Poisson equation in (3.17) must be solved. Equations (3.18)and (3.19) show the boundary conditions, i.e. both the electric field and the potentialmust be null at the extremity of the depletion region, located at r = rd.

1

r

∂

∂rr

∂

∂rV (r) =

qN

εs

, (3.17)

∂V (r)

∂r

∣∣∣∣r=rd

= 0, (3.18)

V (rd) = 0. (3.19)

The density of carriers N is equal to the doping of the semiconductor and can be greaterthan n0, εs is the dielectric permittivity of the semiconductor. The solution for the Poissonequation is shown in (3.20).

V (r) =qN

4εs

(r2 − r2

d

)− qNr2

d

2εs

(ln r − ln rd.) (3.20)

The expression for the electrostatic potential V (r) is different from its planar analo-gous, here named Vp(x), which only depends on the distance x from the interface betweenthe insulator and the semiconductor and, as reported by [Sze81], is given by

Vp(x) =qN

2εs

(x−W )2 . (3.21)

3.3. SATURATION REGION 33

The potential Vs at the interface between the insulator and the semiconductor is

Vs = V (ri) =qN

4εs

(r2i − r2

d

)− qNr2

d

2εs

(ln ri − ln rd) (3.22)

and (3.20) can be written again as

V (r) = Vs

[1− qN

4εs

(r2i − r2

)+

qNr2d

2εs

(ln ri − ln r)

]. (3.23)

Since (3.16) holds, the potential Vs becomes

Vs =qN

4εs

(−2Wri −W 2

)+

qN (r2i + 2Wri + W 2)

2εs

ln

(1 +

W

ri

). (3.24)

To go further in the development of the model one must proceed with some approxima-tions: if one supposes W ri, which is reasonable for the devices developed so far, (3.24)can be expanded in a Taylor series up to the second order to obtain

Vs =qN

2εs

W 2, (3.25)

which is also the expression for the potential at the interface semiconductor/insulator fora planar FET.

The voltage drop on the insulator Vi is given by

Vi =Qs

Ci

=qN

2riCi

(r2d − r2

i

)=

qN

2riCi

(W 2 + 2riW

), (3.26)

where Qs is the charge per area unit in the depletion region.In order to find the width of the depletion region W , we need to solve

− Vg − Vfb + V (z) = Vi + Vs. (3.27)

An analytical solution can be found under the same hypotheses of (3.25), to obtain

W =

− 1Ci

+

√1

C2i

+(

1εs

+ 1riCi

)2[−Vg−Vfb+V (z)]

qN

1εs

+ 1riCi

. (3.28)

The differences with the planar analogous are in the term 1/riCi, which is not present, andin the expression for the capacitance for area unit, which is a function of the gate radiusand the insulator thickness. Fig. 3.3 shows the width W of the depletion region versusthe insulator thickness di. For a cylindrical device, W is always smaller than the width ofa planar device with the same thickness. Both widths become null as the thickness tendsto infinity.


0 200 400 600 800 1000 12000

50

100

150

200

250

300

Insulator thickness (nm)

Dep

letio

n re

gion

wid

th (

nm)

CylindricalPlanar

Figure 3.3: Depletion region vs. insulator thickness. A planar device and a cylindricaldevice, but with rg = 1 µm, are shown. −Vg + V (z) = 10 V, Vfb = 0.

Figure 3.4 shows the width of the depletion region as a function of the gate radius. Asfor the threshold voltage, it tends asymptotically to the width of a planar TFT. Figure3.5 shows the behavior of the TFT. For those values of z where there is no depletionregion, the value of dR is still given by (3.9) and the accumulation layer extends from thesource up to where V (za) = Vg; instead, for those values of z where the depletion region ispresent, only the free carriers within the volume which has not been depleted (i.e. betweenradius rd and rs) contribute to the current. In this case, the resulting elemental resistanceis given by

dR =dz

qµn0π (r2s − r2

d). (3.29)

Thus,

dV = IddR =Iddz

qµn0π (r2s − r2

d). (3.30)

Differentiating W with respect to V gives

dW =dV

qN[(

1εs

+ 1riCi

)W + 1

Ci

] . (3.31)

3.3. SATURATION REGION 35

0 20 40 60 80 10026

27

28

29

30

31

32

33

Gate radius (μm)

W (

nm)

Planar value

Figure 3.4: Width of the depletion region vs. gate radius for −Vg+V (z) = 10 V, Vfb = 0 V,di = 500 nm. The depletion region for a planar TFT with the same insulator characteris-tics is equal to W = 32.15 nm.

L

W(z)ds

za

n0

Contact Contact

Accumulationlayer

0 z

Figure 3.5: Section of the TFT transistor. The depletion region at the lower right cornerhas a width W (z). The accumulation layer extends up to za.

Substituting (3.31) and (3.16) into (3.30) leads to

dW =Iddz

q2Nµn0π[(

1εs

+ 1riCi

)W + 1

Ci

] [r2s − (ri + W )2] . (3.32)

As in [HHB+98] [BJDM97], to obtain the saturation current Idsat, one must sum thecontribution given by the integration of (3.10) up to V (za) = Vg, with that given by the


integration of (3.30), between za and the drain contact, so

Idsat =Z

LµCi

∫ Vg

0

(Vg − Vth − V ) dV +1

L

∫ Vd

Vg

qµn0π(r2s − r2

d

)dV. (3.33)

Changing the integration variable in the second addend in (3.33) from V to W , withW (Vg) = 0 and W (Vd) = rs − ri = ds, leads to

Idsat =Z

LµCi

∫ Vg

0

(Vg − Vth − V ) dV +

+1

L

∫ ds

0

qµn0π[r2s − (ri + W )2] qN [( 1

εs

+1

riCi

)W +

1

Ci

]dW

. (3.34)

Integrating:

Idsat =Z

LµCi

(V 2

g

2− VthVg

)+

+q2µn0Nπ

L

ds

Ci

[(r2s − r2

i

)− dsri −

d2s

3

]+

+d2

s

2

(1

εs

+1

riCi

)[(r2s − r2

i

)− 4dsri

3− d2

s

2

]. (3.35)

The pinch-off voltage Vp, i.e. the voltage for which the depletion region extension is equalto ds, can be obtained substituting Vp = Vg − V (z) in (3.28). This leads to

Vp = ±qN

[d2

s

2

(1

εs

+1

riCi

)+

ds

Ci

]. (3.36)

The saturation current becomes:

Idsat =Z

LµCi

(V 2

g

2− VthVg

)+

Z

LµCi

VpVth

2+

+qµn0π

L

[−Vpdsri − Vp

d2s

2− qNd3

sri

6εs

]. (3.37)

Under the hypotheses that n0 = N and ri ds, the threshold voltage is equal to thepinch-off voltage, Vp = Vth and all the terms in the second row of (3.37) can be neglected,to obtain the equation:

Idsat =Z

2LµCi (Vg − Vth)

2 . (3.38)

Figure 3.6 shows a simulation of the Id − Vd characteristics of a cylindrical deviceand a planar device, with same insulator and semiconductor thicknesses. The differentbehavior is primarily due to the different threshold voltages of the devices. Increasing thegate radius as well as reducing the insulator thickness would cause the curves to be moresimilar.

3.4. EXPERIMENTAL RESULTS 37

−100 −80 −60 −40 −20 0−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

−6

Vd (V)

Id (

A)

Cylindric rg = 1 μm

Cylindric rg = 10 μm

Planar

Figure 3.6: Id − Vd characteristics simulated for cylindrical devices with rg = 1 µm andrg = 10 µm and a planar device. Both have Vfb = 0 V, di = 500 nm, ds = 20 nm. It issupposed that the devices have Z/L = 1. Characteristics are simulated for a variation ofthe gate voltage from 0 V to −100 V in steps of −20 V.

3.4 Experimental results

Cylindrical transistors have been realized by means of high vacuum evaporation of pen-tacene on fibers with an inner core of copper, covered by an insulating layer of poly-imide. In particular, the gate of the developed structure is a copper cylinder with radiusrg = 22.5 µm and is uniformly surrounded by a polyimide insulating layer with a thicknessdi = 500 nm. The organic semiconductor, pentacene, was evaporated over the polyimide,with an estimated thickness ds = 50 nm. Since, during the evaporation process, the wirewas not rotated with respect to the crucible, only about half of the device was uniformlycovered with the semiconductor, therefore the width of the conducting channel Z is alsohalf of the one reported in (3.8).

Source and drain contacts have been realized with gold as well as a conductive poly-mer, Poly(ethylene-dioxythiophene)/PolyStyrene Sulfonate (PEDOT:PSS), by means ofhigh vacuum evaporation and soft lithography, respectively. Figure 4.11 shows the Id−Vg

curves of the experimental data, measured by a HP4155 semiconductor parameter ana-


0 20 40 60 80 10010

−10

10−9

10−8

10−7

10−6

10−5

Vg (V)

Id (

A)

RealFitted

PEDOT

Gold

Figure 3.7: Id− Vg characteristics for devices with gold and PEDOT contacts. The drainvoltage is kept at Vd = −100 V.

lyzer, together with the ones of the fitted data, whereas fig. 3.9 and 3.10 show the Id−Vd

curves.To give an estimate of the electrical parameters of the devices, a least squares fitting

of the experimental data with equations (3.15) and (3.37) was performed. In sections2.4.1 and 2.4.3 it has already been pointed out that in organic devices the mobility candepend on the electric field determined by the gate bias. A semi-empirical power law wasfound for both the variable range hopping and the multiple trap and release model:

µ = κ (Vg − Vth)α . (3.39)

For the MTR model, for example, a p-type transistor would have the following fielddependent mobility:

µ = µ0NV

Nt0

(Ci (Vg − Vth)

qNt0

)TcT−1

, (3.40)

where Nt0 is the total surface density of traps, Tc is a characteristic temperature andNV is the effective density of states at the band edge. Equation (3.40) shows the samegate-voltage dependency of (3.39) and can be heuristically extended to cylindrical TFTs,provided that (3.4) and (3.12) are used as capacitance per area unit and threshold voltage,

3.4. EXPERIMENTAL RESULTS 39

0 20 40 60 80 1000.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Vg (V)

μ (c

m2 V

−1 s−

1 )

PEDOT

Gold

Figure 3.8: Mobility vs. gate voltage. The drain voltage is kept at Vd = −100 V.

respectively. It is worth noting that, since the mobility only depends on the gate voltage,the model developed in section 3.2 and 3.3, where the integrations are performed withrespect only to drain voltage, still holds. A more rigorous derivation of the field dependentmobility for cylindrical transistors would require the resolution of the Poisson equationand the current equation for the device in the accumulation region considering a field-dependent conductance.

To give an estimate of the threshold voltage, Id − Vg data have been fitted withequations (3.37), together with (3.39) which accounts for the field dependent mobility.Moreover, to determine possible deviances from (3.39), the mobility has been directlyextracted from the transfer characteristic considering the previously estimated thresholdvoltage and is shown in figure 4.14. Finally the contact resistance, which could also befield dependent [NSGJ03], has been extracted from the output curves. However, it wassupposed to be constant not to introduce too many independent parameters in the fittingprocedure. The extracted parameters are reported in table 3.4.


−70 −60 −50 −40 −30 −20 −10 0−8

−7

−6

−5

−4

−3

−2

−1

0

1x 10

−7

Vd (V)

Id (

A)

Real dataFitted data

Figure 3.9: Id − Vd characteristic for device with gold contacts. The gate voltage is keptat Vg = −100 V.

Device Vth (V) κ α Rs (MΩ) Ion/Ioff

Au 5.1 1.9 · 10−3 0.66 7.4 7 · 103

PEDOT 13.8 0.96 · 10−3 0.85 2.3 3 · 103

Table 3.2: Electrical parameters of the devices.

3.5 Summary

In this chapter we have developed a model which describes the electrical characteristicsof TFTs with cylindrical geometry. Under reasonable approximations (i.e insulator andsemiconductor thicknesses are much smaller than the gate radius), the model becomes verysimilar to its known planar analogous, provided that parameters like channel width andcapacitance per area unit are changed consistently. Differences between the two geometriesbecome more important as the gate radius is reduced. The insulator thickness too canhave an important role in the determination of the threshold voltage and the width of thedepletion region, leading to significative differences in the output curves. Experimentalresults have been fitted with the proposed model, showing that it can adequately describethe physical behavior and the electrical characteristics of the cylindrical devices. Future

3.5. SUMMARY 41

−70 −60 −50 −40 −30 −20 −10 0−16

−14

−12

−10

−8

−6

−4

−2

0

2x 10

−7

Vd (V)

Id (

A)


Figure 3.10: Id − Vd characteristic for device with PEDOT contacts. The gate voltage iskept at Vg = −100 V.

work in this topic includes, in the theoretical analysis, a rigorous determination of thecurrent equations when a field dependent mobility given by MTR or VRH is considered; onthe experimental side, the development of simple circuits with interconnected transistorsis the future step for the development of the e-textile.


Chapter 4

Short channel effects

In the previous chapter theoretical models for organic transistors have been extended tocylindrical geometries, but no hypotheses on the length of the channel L were made. How-ever, channel length in organic thin film transistors is an important parameter, because itcan lead to substantial deviations when the physical dimensions of the device scale down.Considerations on this point are reported, for example, by Torsi et al. [TDK95], whileexperimental evidence by Chabinyc et al. [CLS+04] shows that, already for devices withchannel length L < 10 µm, short channel effects arise.

The equations (2.9) and (2.23),introduced in section 2.3.1, cannot describe completelythe device characteristics, as they exhibit several non-idealities, as reported by Haddocket al. [HZZ+06]. To account for these short channel effects, like space charge limitedcurrents or the mobility dependence on the longitudinal electric field (source-drain), moresophisticated models have been proposed. For instance, it has been recently pointed outby Koehler and Biaggio [KB04] that, for high applied voltages and short channel lengths,the current conduction can be significantly influenced by space charge limited current(SCLC) phenomena, i.e. the carriers injected from the contacts can constitute a spacecharge region which affects the behavior of the device. SCLC conduction can explain thenon-saturation of the output curves for high drain voltages in OTFTs with short channellengths. Experimental evidence of SCLC effects has been reported in [CLS+04].

The mobility dependence on the longitudinal electric field (source-drain) Ex can bedescribed by the Poole-Frenkel model described in section 2.4.4:

µ(Ex) = µ0 exp(γ√Ex

)(4.1)

where µ0 is the zero-field mobility and γ is a prefactor which depends on the materialand is generally inverse in proportion to the temperature T , at least for T > 50 K [HN07].Stallinga et al. [SGB+04] [SG06b] show that Poole-Frenkel mobility can be responsible ofthe non-linearities often observed in the output curves for low drain voltages. Hamadani

43

44 CHAPTER 4. SHORT CHANNEL EFFECTS

et al. [HN07] [HRG+07] report field dependent mobilities in OTFTs which are consistentwith the Poole-Frenkel model for different temperatures. Numerical simulations have beencarried on by Bolognesi et al. [BCLC03] [BBM+04] including Poole Frenkel mobility inthe drift-diffusion model, showing good agreement with experimental results.

In this chapter we will develop a model which accounts for SCLC conduction togetherwith a field dependent mobility which is described by the Poole-Frenkel model, determin-ing the device characteristic equations at different applied biases.

In section 4.1 a description of the hypotheses on the devices and their geometry areshown. In section 4.2 the model is obtained for the linear, depletion and saturationregimes. In section 4.3, the model is validated with experimental results obtained fromshort channel OTFTs.

4.1 The device

We consider a device as reported in fig. 4.1, which is also the device already describedin section 2.3.1. It is a bottom contact OTFT, with insulator thickness, dielectric per-mittivity and capacitance per unit area equal to di, εi and Ci, respectively, and withsemiconductor thickness, dielectric permittivity and capacitance per unit area equal tods, εs and Cs, respectively. The channel length is equal to L, whereas its width is Z. Itis assumed that the semiconductor has a free density of carriers n0 which is equal to thedoping of the semiconductor N . The contacts are assumed to be ohmic. Nevertheless, itshould be noted that real OTFTs behavior can be deeply affected by source-drain contactresistances, especially in short channel devices. In section 4.3 experimental data will beanalyzed extending the model to include also contact resistances. A complete list of the

ds

di

L

Source

Gate

Insulator

Semiconductor Draindx

0

y

Figure 4.1: Geometry of a bottom contact OTFT.

parameters used in this chapter, together with the values used for simulations, is reportedin table 4.1.

4.2. MODEL DERIVATION 45


Device width Z 5 mm

Device length L 15 µm

Insulator thickness di 200 nm

Insulator dielectric permittivity εi 3.9 · ε0

Semiconductor thickness ds 100 nm

Semiconductor dielectric permittivity εs 3.9 · ε0

Semiconductor doping N 1014 cm−3

Free carrier density n0 1014 cm−3

Free carrier mobility µ0 10−4 cm2/Vs

Table 4.1: Simulation parameters.

4.2 Model derivation

a) Accumulation

b) Depletion

c) SCLC regime

xp

n0

Accumulationlayer

Source DrainSCLC

x0

0 x

Source Semiconductor (n )0 Drain

Accumulationlayer

W(x)

x0

n0

Accumulationlayer

Source Drain

Figure 4.2: Modes of operation of the OTFT. The origin of the x-axis is at the sourcecontact.

The first issue when describing the behavior of an organic thin-film transistor is inthe definition of the threshold voltage, which can be expressed as the voltage to apply


to the gate electrode, referenced to the source, in order to create the accumulation layer.This definition implies that an ideal accumulation device, where there are no localizedstates originating from acceptors or donors, has a null threshold voltage [SG06a]. Withoutspeculating on the possible origins of non-ideal values of the threshold voltage, we willassume in the following that it has a generic value of Vth. In the following paragraphs, wewill assume to have an n-type transistor; for a p-type transistor the sign of the currentsand applied voltages have to be changed accordingly.

4.2.1 Linear region

The characteristic equation of the device in the linear regime (fig. 4.2-a), i.e. for Vd <

Vg − Vth, can be obtained considering that the total drain current Id can be expressed asthe sum of the bulk current originating from the free carriers in the semiconductor andthe current determined by the carriers in the accumulation layer. Therefore

Id

Zµ(Ex)= − [Qg(x) + Q0]

dV

dx, (4.2)

where Q0 is the surface density of the free carriers given by

Q0 = ±qn0ds, (4.3)

wherein the sign of the right hand side of (4.3) is that of charge of the carriers (plus forholes, minus for electrons). The surface density of the carriers in the accumulation layerQg(x) which forms as a capacitive effect, is

Qg(x) = −Ci [Vg − Vth − Vs(x)− V (x)] , (4.4)

wherein Vs(x) is the ohmic drop in the bulk of the semiconductor, which can be generallyneglected [HHK98], and V (x) is the potential with respect to the source at the coordinatex. Then we can state that

Id

Zµ(Ex)= Ci [Vg − Vth + V0 − V (x)]

dV

dx, (4.5)

where V0 = −Q0/Ci. Under the gradual channel approximation (i.e. the electric fieldalong x is negligible with respect to that along y), V (x) increases from 0 at the source toVd at the drain and also ∣∣∣∣dEy

dy

∣∣∣∣ ∣∣∣∣dEx

dx

∣∣∣∣ . (4.6)

Moreover, the electric field along x can be expressed as

Ex 'Vd

L, (4.7)


so the mobility is constant along the channel and

µ(Ex) ' µ0 exp

(γ

√Vd

L

). (4.8)

Integration of (4.5) between V (0) = 0 and V (L) = Vd gives

Id =Z

Lµ0 exp

(γ

√Vd

L

)Ci

[(Vg − Vth + V0) Vd −

V 2d

2

]. (4.9)

4.2.2 Depletion region

For Vd > Vg−Vth, a depletion region forms, starting from a point x0 where V (x0) = Vg−Vth

and ending at the drain contact (fig. 4.2-b). Solving the Poisson equation along the y

direction [HHK98] allows to determine its extension, which is

W (x) =εs

Ci

√

1 +2C2

i [V (x)− (Vg − Vth)]

qNεs

− 1

. (4.10)

The condition W = ds is verified at the pinch-off abscissa xp and the pinch-off voltagecan be obtained from W (Vp) = ds, so that

Vp = (Vg − Vth) + V0

(1 +

Ci

2Cs

). (4.11)

For every x0 ≤ x ≤ xp contributions to the current are only given by the free carriersin the volume which has not been depleted, so

Id

Zµ(Ex)= qn0 [ds −W (x)]

dV

dx. (4.12)

We consider the electric field along x constant in this region, up to the pinch-offabscissa xp, so again the mobility can be expressed as in (4.8). It should be noted thatthis approximation is used only in a limited range of x since, for thin film transistors, thepinch-off voltage Vp is near to (Vg − Vth) and, in the limit of ds → 0, Vp = (Vg − Vth) andx0 ≡ xp. Integrating (4.12) between V (x0) = (Vg − Vth) and V (L) = Vd and applying thecontinuity of the current with (4.9) at Vd = (Vg − Vth) we find

Id =Z

Lµ0 exp

(γ

√Vd

L

)Ci [(Vg − Vth) V0+

+(Vg − Vth)

2

2

]+ (Ci + Cs) V0 [Vd − (Vg − Vth)] +

+(CsV0)

2

3Ci

[1−

(1 +

2Ci

V0Cs

[Vd − (Vg − Vth)]

)3/2]

. (4.13)


4.2.3 SCLC region

Beyond the pinch-off abscissa the gradual channel approximation is no longer valid, theelectric field varies along the channel and the exact determination of the current inthe semiconductor would require the numerical resolution of the two dimensional drift-diffusion model. However, since in the pinch-off region the effects of the drain potentialbecome dominant compared to the effects of the gate potential, we can assume that thebehavior of the device beyond the pinch-off resembles that of a thin film SCLC region.Grinberg et al. [GLPS89] determined the expression for the SCLC current in a thin filmwith ohmic contacts, but now a field-dependent mobility must also be considered. To finda simple yet useful solution, we first reproduce the case study the authors simulated, i.e. athin film n+−i−n+ structure with strip ohmic contacts like the one reported in figure 4.3.The device is surrounded by vacuum, as the electric field related to the charge in the thin

in+ n+

0 l x

Vacuumdv

dv Vacuum

Figure 4.3: Geometry of the thin film n+ − i − n+ structure: the device has a length ofl = 4.8 µm and it has a strip contact geometry, with two 25 µm long ohmic contacts. Thestructure is surrounded by dv = 10 µm of vacuum.

film is supposed to exceed the boundaries of the structure. This is an important point,because it also means that a one dimensional model can only approximately describe thebehavior of a thin film structure.

A finite element drift-diffusion simulation of the structure is performed using thecommercial software Sentaurus by Synopsis [sen]. Poole-Frenkel mobility is implementedand, for various values of γ, I−V curves were determined. The output curves are reportedin figure 4.4. The following law is found to provide good approximations to the simulatedcurves, introducing a fitting parameter β which depends on γ. It is similar to the onefound in [GLPS89], but has an additional factor which accounts for the field dependent


−100 −80 −60 −40 −20 010

−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

NumericalApproximation

γ = 103 (m/V)1/2

γ = 102 (m/V)1/2

γ = 104 (m/V)1/2

Figure 4.4: I − V curves for n+ − i− n+ structure for different γ values.

mobility.

I = 0.6Zεsµ0V 2

l2exp

(β · γ

√V

l

). (4.14)

The prefactor in the current is kept equal to the one reported by Grinberg et. al. fora structure with strip contacts. Extracted values of β values are reported in table 4.2.Equation (4.14) provides worse approximations of the numerical simulation as the γ pa-rameter increases, because the dependence on the exponential of the square root of thefield increases with γ.

γ (m/V)1/2 β

104 0.72103 0.63102 0.14

Table 4.2: Extracted fitting parameter β.

Figures 4.5 and 4.6 show the absolute value of the electric field, which clearly exceedsthe boundaries of the device.


X

Y

0 10 20 30 40 50

-10

-5

0

5

10

ElectricField

1.6E+05

1.3E+05

9.7E+04

6.5E+04

3.3E+04

3.9E+02

Figure 4.5: Absolute value of the electric field in the structure and in vacuum.

Since l = (L− xp) and Vx = (Vd − Vp) we obtain

Id (L− xp) = Z0.6µ0εs(Vd − Vp)

2

(L− xp)exp

(βγ

√(Vd − Vp)

(L− xp)

). (4.15)

The Idxp term in the left hand side can be determined applying the continuity with(4.13) at L = xp, therefore we find

Id =Z

L

Ci

2

(((Vg − Vth) + V0)

2 +V 2

0 Ci

3Cs

)µ (Vp) +

+ 0.6µ0εs(Vd − Vp)

2

(L− xp)exp

[βγ

√Vd − Vp

(L− xp)

], (4.16)

wherein µ (Vp) = µ0 exp(γ√

Vp/xp

)is the mobility at the pinch-off point and contin-

ues to increase as the point becomes closer to the source contact.The pinch-off abscissa xp can be determined substituting the term Id found in (4.24)

back in (4.23). This leads to


X

Y

24 26 28 30

-0.4

-0.2

0

0.2

0.4

ElectricField

2.5E+05

2.0E+05

1.5E+05

1.0E+05

5.1E+04

3.9E+02

Figure 4.6: Zoom of the absolute value of the electric field in the structure and in vacuum.The structure begins at x = 25 µm and ends at x = 29.8 µm.

Ciµ (Vp)

[(Vg − Vth + V0)

2 +V 2

0 Ci

3Cs

](L− xp)

2

= 2 · 0.6xpµ0εs (Vd − Vp)2 exp

[βγ

√Vd − Vp

(L− xp)

]. (4.17)

In fig. 4.7 equation (4.25) is solved as a function of the drain voltage Vd for severalgate voltages Vg. For Vd < Vp the depletion region does not extend over the wholesemiconductor thickness ds, whereas for Vd > Vp the pinch-off point gradually approachesthe source contact.

Output curves simulated for devices with high and low γ are shown in fig. 4.8 and4.9, respectively. In fig. 4.8, the curves exhibit a super-linear behavior for Vd < Vg − Vth,which is due to the Poole-Frenkel mobility, whereas SCLC effects, for Vd > Vp, result inthe non-saturation of the curves. When the mobility dependence on the electric field Ex islow, that is when γ is low enough to make the argument of the exponential in the mobilitytend to zero, as in fig. 4.9, the output curves resemble more those of the conventional


−80 −70 −60 −50 −40 −30 −20 −10 05

6

7

8

9

10

11

12

13

14

15

Vd (V)

x p (μm

)

Vg = −10 V

Vg = −30 V

Vg = −50 V

Vg = −70 V

Figure 4.7: Pinch-off abscissa xp as a function of the drain voltage Vd with L = 15 µm,γ = 1.5 · 10−4 (m/V)1/2 and Vth = 0. Vg = −10,−30, . . . ,−90 V. For Vg = −90 V thedevice remains in accumulation regime, since Vd < Vg, so xp = L.


OTFT quadratic model. Even the non-saturation of the curves is less evident, for theSCLC is not enhanced by the Poole-Frenkel mobility.

−80 −70 −60 −50 −40 −30 −20 −10 0−8

−7

−6

−5

−4

−3

−2

−1

0

1x 10

−5

Vd (V)

I d (A

)

Vg = −90 V

Vg = −70 V

Vg = −50 V

Vg = −30 V

Vg = −10 V

Figure 4.8: Output characteristics for a p-device with L = 15 µm, γ = 1.5 ·10−3 (m/V)1/2

and Vth = 0. Vg = −10,−30, . . . ,−90 V.

To better show how the channel length can affect the electrical characteristics of thedevices, fig. 4.10 shows the output curves for different channel lengths L and the sameapplied gate voltage Vg. As the channel length decreases both Poole-Frenkel mobility andSCLC effects become more evident, leading to devices which do not saturate at all.

One last remark is about the doping value used in our calculations, which determinesthe potential V0. With a doping N = 1014 cm−3, V0 = 9.2 mV. For typical doping valuesof P3HT [MMH+01], which are equal or less N = 1016 cm−3, the potential V0 still remainsnegligible (V0 = 0.92 V) when compared to the voltages applied to the electrodes. Thishappens because V0 is directly proportional to the thickness of the thin film ds.

4.2.4 The limit of thick film

When the semiconductor thickness is large enough so that no field exceeds the boundariesof the structures, the proposed model cannot predict correctly the current. In this case, weassume that beyond the pinch-off abscissa the effects of the longitudinal field Ex become


−80 −70 −60 −50 −40 −30 −20 −10 0−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−6

Vd (V)

I d (A

)

Vg = −90 V

Vg = −70 V

Vg = −50 V

Vg = −30 V

Vg = −10 V

Figure 4.9: Output characteristics for a p-device with L = 15 µm, γ = 1.5 ·10−4 (m/V)1/2

and Vth = 0. Vg = −10,−30, . . . ,−90 V.


−80 −70 −60 −50 −40 −30 −20 −10 0−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0x 10

−5

Vd (V)

I d (A

)

L = 5 μm

L = 3 μm

L = 7, 9, ..., 13 μm

Figure 4.10: Output characteristics for p-devices with L = 3, 5, . . . , 13 µm. γ = 1.5 ·10−4 (m/V)1/2. Vg = −25 V.


dominant compared to those of Ey, so that the following approximation can be made:∣∣∣∣dEx

dx

∣∣∣∣ ∣∣∣∣dEy

dy

∣∣∣∣ . (4.18)

The Poisson equation becomesd2V

dx2= −qn

εs

, (4.19)

where n is the volume charge density of the carriers. The model is one dimensional. Thecurrent is

Id

Z= dsqnµ (Ex) Ex. (4.20)

The mobility cannot be approximated with (4.8), since the electric field along the x

direction is no more constant due to the space charge present. Combining (4.19) with(4.20) we obtain

Id

Z= εsdsµ0 exp

(γ√Ex

)Ex

dEx

dx. (4.21)

Equation (4.21) cannot be solved analytically, however the Murgatroyd approach [?] canbe followed to determine the SCLC current in the semiconductor. For a semiconductor oflength l and applied voltage Vx, an approximated solution for (4.21) can be found:

Id = Z9

8µ0εsds

V 2x

l3exp

(0.891

kTγ

√Vx

l

). (4.22)

Since l = (L− xp) and Vx = (Vd − Vp) we obtain

Id (L− xp) = Z9

8µ0εsds

(Vd − Vp)2

(L− xp)2 exp

[0.891

kTγ

√(Vd − Vp)

(L− xp)

]. (4.23)

The Idxp term in the left hand side can be determined applying the continuity with (4.13)at L = xp, therefore we find

Id =Z

L

Ci

2

[(Vg − Vth + V0)

2 +V 2

0 Ci

3Cs

]µ (Vp) +

+9

8µ0εsds

(Vd − Vp)2

(L− xp)2 exp

[0.891

kTγ

√Vd − Vp

(L− xp)

], (4.24)

wherein µ (Vp) = µ0 exp(γ√

Vp/xp

)is the mobility at the pinch-off point and contin-

ues to increase as the point becomes closer to the source contact.The pinch-off abscissa xp can be determined substituting the term Id found in (4.24)

back in (4.23). This leads to

4.3. EXPERIMENTAL RESULTS AND PARAMETER EXTRACTION 57

4Ciµ (Vp)

[(Vg − Vth + V0)

2 +V 2

0 Ci

3Cs

](L− xp)

3

= 9xpµ0dsεs (Vd − Vp)2 exp

[0.891

kTγ

√Vd − Vp

(L− xp)

]. (4.25)

4.3 Experimental results and parameter extraction

To perform a comparison between the developed model and real data, bottom gate shortchannel OTFTs with P3HT semiconductor have been realized, with channel length L =

5 µm and L = 2.5 µm. They have been produced by spin-casting the pristine polymersolution onto a 230 nm SiO2 dielectric, thermally grown on a N+Si wafer, used as gateelectrode. Sputtered gold Source and Drain electrodes were used to contact the polymer.Before coating the active layer the SiO2 surface was treated with a hexamethyldisilazane(HMDS) primer, to improve the film adhesion and formation. An interdigitated layout,obtained by means of conventional photolithography, was used for the OFET where Z =

10 mm is the FET channel width. The dielectric capacitance per unit area is Cins =

1.5 · 10−4 F/m2. The dielectric constant used in the case of the P3HT is εs = 2 aswe determined by means of quasi static C-V measurements. The nominal value of thesemiconductor thickness is 100 nm.

Two sets of measurements were performed: transfer curves are reported in fig. 4.11and output curves are shown in fig. 4.12 and 4.13.

To extract the electrical parameters of the devices two contact resistances Rs, at boththe source and drain contacts, were added to the model [NFS07], substituting Vd with(Vd − 2RsId) and Vg with (Vg −RsId) into the equations used for the fitting, namelyequations (4.9), (4.13), (4.24) and (4.25). To reduce the number of fitting parameters, wesupposed to have a very low density of free carriers, so that one can assume V0 ' 0 V.

Fitting of the transfer curves, which is reported in fig. 4.11, allows the determinationof the threshold voltage Vth, the prefactor γ, the zero-field mobility µ0 and the seriesresistance Rs. Fitting values are reported in table 4.3.

For low values of Vg, the fitting slightly deviates from the experimental data and thiscould be due to the zero-field mobility which is considered constant, whereas it coulddepend on Vg [TMBdL03]. Therefore, to account for possible dependencies on the gatevoltage, the mobility µ0 has also been extracted from each of the output curves and isshown in fig. 4.14. It can be observed that its magnitude is slightly different from thatextracted from the transfer curves, and this is consistent with the higher currents in the


-80 -70 -60 -50 -40 -30 -20 -10 0-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1x 10

-4

Vg (V)

Id(A

)Fitted data

Real data

L = 5 m

L = 2.5 m

-80 -70 -60 -50 -40 -30 -20 -10 0

-10-3

-10-4

-10-5

-10-6

-10-7

Vg (V)

Id(A

) L = 5 m

L = 2.5 m

Figure 4.11: Id − Vg characteristics for L = 5 µm and L = 2.5 µm devices. Drain voltageis kept at Vd = −75 V.

transfer curves measured for the same applied voltages. Also, µ0 increases with the gatevoltage. It should be noted that the small differences between the fitting parameters ofthe device with L = 2.5 µm and the device with L = 5 µm, which represent physicalproperties of the used materials, can be ascribed to differences in the realization process.

Vth (V) µ0 (cm2/Vs) γ (m/V)1/2 Rs (kΩ)

L = 2.5 µm -2.8 3.6 · 10−4 5.8 · 10−4 9

L = 5 µm -7.5 8, 4 · 10−4 4.5 · 10−4 17

Table 4.3: Fitting parameters.

4.4 Conclusion

In this chapter a model which describes the electrical characteristics of OTFTs with shortchannel lengths has been presented. Effects like super-linear output curves for low drainvoltage, as well as non-saturating currents can be adequately described. The model is

4.4. CONCLUSION 59

−80 −70 −60 −50 −40 −30 −20 −10 0−7

−6

−5

−4

−3

−2

−1

0

1x 10

−4

Vd (V)

Id (

A)


Vg = −80 V

Vg = −60 V

Vg = −40 V

Vg = −20 V

Vg = 0 V

Figure 4.12: Id − Vd characteristics for L = 2.5 µm device. Vg = 0,−20, . . . ,−80 V.

−80 −70 −60 −50 −40 −30 −20 −10 0−2.5

−2

−1.5

−1

−0.5

0

0.5x 10

−4

Vd (V)

Id (

A)


Vg = −80 V

Vg = −60 V

Vg = −40 V

Vg = −20 V

Vg = 0 V

Figure 4.13: Id − Vd characteristics for L = 5 µm device. Vg = 0,−20, . . . ,−80 V.


−80 −70 −60 −50 −40 −30 −20 −100

1

2

3

4

5

6x 10

−4

Vg (V)

μ (c

m2 /V

s)L = 5 μmL = 2.5 μm

Figure 4.14: Zero-field mobility as a function of the gate voltage.

based on the assumption that the field dependent mobility of the organic semiconduc-tor follows the Poole-Frenkel theory; moreover, space charge limited currents beyond thepinch-off point are considered. The length of the channel plays a key role in the deter-mination of the electrical characteristics, since the high longitudinal electric field in shortchannel devices increases the carriers mobility and SCLC effects. Experimental results fordevices with 5 µm and 2.5 µm channel lengths have been analyzed and good agreementwith the proposed model is found. Future work in this topic includes an in-depth com-parative analysis between the proposed model and numerical drift-diffusion simulations,where the non saturation is studied considering the thickness of the insulator and there-fore the complete effects of the gate dependent electric field. Moreover, further analysisis required to precisely describe the effects of the contacts.

Chapter 5

Dynamic models for organic TFTs

In this chapter we would like to explore the characteristics of organic transistors as afunction of time. Experimental results in literature, for example the work of Schmecheland von Seggern [SvS04] [SvS02], show that the performances of organic transistors areusually affected by trap states located in the energy gap of the material.

The origin of these localized states is still not completely clear: for example, Kang etal. [KdSFBZ05] report, for vapor-deposited pentacene, that shallow traps can arise fromthe sliding of the molecules along the molecular axis, while two dimensional crystallinity ismaintained. Northrup et. al. discuss the effects of hydrogen related defects in crystallinepentacene, i.e. when the pentacene molecule C22H14 has one more hydrogen atom andbecomes C22H15, but also analyze the effects of oxygen related defects, i.e. when oneoxygen atom replaces one hydrogen atom to give C22H13O. Both defects are originatedby exposure of the semiconductor to air humidity, but hydrogen defects could also comefrom bias stress in the device. Goldmann et al. [GGB06] also refer to water adsorptionon the surface of the insulator, when discussing the generation of discrete traps states insingle-crystal pentacene.

The main macroscopic consequence of the presence of traps in a device is hysteresisin its output, transfer or capacitance-voltage curves [SMS+05] [GKDF05], because trapsare charged and uncharged with specific dynamics which can lead to different operatingconditions, even if the applied bias is the same. Analytical models [LMK07], as well asnumerical simulations [LPS05] [PSH+08], have been proposed to study these phenomena,which still constitute a hot topic in the field of organic electronics. In this chapter, wewill describe how trap recharging can be modeled, and we will show a gallery of severalsimulations, which provide an in-depth view of the effects of the parameters involved intrap dynamics.

61

62 CHAPTER 5. DYNAMIC MODELS FOR ORGANIC TFTS

5.1 Model

The model that describes trap recharging in OFETs is the same that was first introducedby Schockley and Read [SR52] and by Hall [Hal52] and is based on the drift diffusionmodel reported in section 2.5. There, contributions of the trap states to the charge wereincluded in (3.17) as ntr. We will follow the exposition of Sze [Sze81] for its description.

More in detail, two types of traps can be defined, donor traps and acceptor traps.

DONOR traps are uncharged when unoccupied and they carry the charge of one holewhen fully occupied.

ACCEPTOR traps are uncharged when unoccupied and they carry the charge of oneelectron when fully occupied.

A trap is a recombination center, determined by impurities or defects in the semicon-ductor, which can capture or emit one charge carrier. Let us consider the capture of one

Ra

EV

EC

Etr

Rb

Rc

Rd

Figure 5.1: Emission and recombination of carriers for a single level of trap.

electron from the conduction band. For a density of traps Ntr, the density of occupiedcenters is Ntrftr, where ftr is the Fermi distribution function, which gives the probabilitythat a center is occupied by an electron. At the equilibrium it is

F =1

1 + exp [(Etr − EF) /kT ], (5.1)

where Etr is the energy level and EF is the Fermi level. The rate of the capture processis given by

Ra = vth,nσnnNtr (1− ftr) , (5.2)

where the constants vth,n and σn are the carrier thermal velocity and the capture section,respectively. σn gives a measure on how an electron must be near to a center to becaptured. For inorganic devices, the capture sections are about 10−15 cm2 (atomic scale).The emission rate of an electron is given by

Rb = enNtrftr, (5.3)

5.1. MODEL 63

where en is the emission probability of the electron. At the thermal equilibrium thecapture and emission rates must be equal: Ra = Rb. Since at the thermal equilibriumn = ni exp [(EF − Ei) /kT ] the probability of emission becomes

en = vth,nσnni exp [(Etr − Ei) /kT ] . (5.4)

For a trap located near the conduction band, the emission probability becomes greater,as en increases exponentially with Etr − Ei.

Similar considerations lead to the following equations for the capture rate for holes

Rc = vth,pσppNtrftr (5.5)

and the emission rate for holes

Rd = epNtr (1− ftr) . (5.6)

Again, at the thermal equilibrium Rc = Rd, so

ep = vth,pσpni exp [(Ei − Etr) /kT ] . (5.7)

As for the emission probability for electrons, the probability of emission for holes increasesexponentially as the energy level of the trap becomes nearer to the valence band. Thelifetime for holes and electrons, respectively, can be introduced:

τn =1

vth,nσnNtr

, (5.8)

τp =1

vth,pσpNtr

(5.9)

and the balance equation for the net emission-capture rate can be defined:

dntr

dt= Rb −Ra + Rd −Rc. (5.10)

Substituting the expressions for the rates we finally obtain

dntr

dt=

1

τn

[exp

(Etr − Ei)

kTftr − n (1− ftr)

]+

1

τp

[exp

(Ei − Etr)

kT(1− ftr)− pftr

]. (5.11)

Simulations will be performed solving, together with the drift diffusion model, theequation in (5.11). However, in the following section, we also report the most generalmodel, which holds for an arbitrary distribution of traps in the band gap.


5.1.1 An arbitrary density of trapped states

For completeness, a more general mathematical model [GMS07], which accounts for anarbitrary distribution of states, is also presented. Here it is proposed for acceptor traps,but an analogous model can be developed for donor traps changing the sign of the trappedcharges in the Poisson equation (3.17) and substituting p with n and viceversa in all thefollowing equations.

Let’s now consider a general semiconductor with a forbidden energy gap Eg and con-duction band energy EC and valence band energy EV. The constant (in space) density oftrap states in the band gap Ntr is obtained by means of intergration on all the band gapenergies of the energy dependent density of trapped states Mtr(E):

Ntr =

∫ EC

EV

Mtr(E) dE. (5.12)

The position density of the trapped states, which is a function of the position r, the energyE and the time t is given by

ntr(r, E, t) =

∫ EC

EV

Mtr(E)ftr(r, E, t) dE, (5.13)

wherein ftr(r, E, t) is the fraction of occupied trap states for a given energy E. Therecombination rate for electrons and holes can be defined as

Rn =

∫ EC

EV

SnMtr dE, (5.14)

Rp =

∫ EC

EV

SpMtr dE, (5.15)

where

Sn =1

τnNtr

[n0ftr − n (1− ftr)] , (5.16)

Sp =1

τpNtr

[p0 (1− ftr)− pftr] . (5.17)

Trap dynamics is determined by the balance equation, which gives an additional differen-tial equation

∂ftr

∂t= Sp − Sn =

p0

τpNtr

+n0

τnNtr

− ftr

(p + p0

τpNtr

+n + n0

τnNtr

), (5.18)

which is coupled to the continuity equations (2.64) and (2.65), which include the recom-bination rates defined by (5.14) and (5.15). The rate constants are τn(E), τp(E), n0(E),

5.2. SIMULATIONS 65

p0(E) and the law of mass action n0(E)p0(E) = n2i holds, where ni is the independent

intrinsic concentration. Integrating (5.18) leads to

∂ntr

∂t= Rp −Rn. (5.19)

For a single level of traps located at energy Etr the energy dependent density of trappedstates becomes

Mtr = Ntrδ (E − Etr) . (5.20)

It follows that the recombination rates for electrons and holes become

Rn =1

τn

[n0

ntr

Ntr

− n

(1− ntr

Ntr

)](5.21)

Rp =1

τp

[p0

(1− ntr

Ntr

)− p

ntr

Ntr

]. (5.22)

For a steady-state solution (∂ftr/∂t → 0) the fraction of occupied trap states becomes

ftr =τnp0 + τpn

τn(p + p0) + τp(n + n0). (5.23)

The recombination rate for electrons equals the hole recombination rate and

Rn = Rp = R(n, p) =(n2

i − np) ∫ 1

0

Mtr(E)

τn(E) [p + p0(E)] + τp(E) [n + n0(E)]dE. (5.24)

For one trap level the well known Schokley-Read-Hall model is found:

R(n, p) =n2

i − np

τn(p + p0) + τp(n + n0). (5.25)

5.2 Simulations

In order to understand the effects of the different physical parameters, several simulationshave been performed on two typical case studies, i.e. a bottom contact transistor and ametal-insulator -semiconductor (MIS) structure. The structures are reported in figure 5.2and 5.3, whereas geometrical parameters are reported in table 5.1.

Material parameters are specified in table 5.2. These values are typical for a semi-conductor like pentacene [JL04]. The metal workfunction for the gate contact is set toφm,gate = 0 eV whereas source and drain contacts, as well as the bulk contact for the MISstructure, have φm = 5 eV. In this way the contacts are ohmic, as φm > φs. Temperatureis T = 300 K. Densities NC and NV are assumed to be twice (spin up and down) thedensity of molecules of pentacene. Unless specified, the doping profile is assumed to beNA = 1016 cm−3. This implies ni =

√NCNV exp (−Eg/2kT ) = 3.695 cm−3.


ds

di

L

Source

Gate

Insulator

Semiconductor Drain

Figure 5.2: Geometry of a bottom contact OTFT.

L

ds

di

Body contact

Gate

Insulator

Semiconductor

Figure 5.3: Geometry of a MIS.


Channel length L 75 µm

Channel width W 5 mm

Dielectric thickness di 1.6 µm

Semiconductor thickness ds 50 nm

Table 5.1: Geometry parameters for MIS and bottom contact OTFT.

The parameters for the trap distribution are chosen from table 5.3: the degrees offreedom are the density of trap and the product between the cross section and the thermalvelocity. Simulations will be performed for an energy level which usually lies 0.3 eV overthe valence band or under the conduction band (deep trap). Traps are physically locatedat the interface between the insulator and the semiconductor. Here they are most effective,but they can also be spatially distributed in the semiconductor bulk.

Transient simulations are performed for the gate sweep in table 5.4. For trans-characteristic curves, the drain voltage is first set at Vd = −25 V using a stationary

5.2. SIMULATIONS 67


Dielectric permittivity εi 4

Semiconductor permittivity εs 4

Electron affinity qχ 2.5 eV

Energy gap Eg 2.5 eV

Valence band density NC 2.8 · 1021 cm−3

Conduction band density NV 2.8 · 1021 cm−3

Electrons mobility µn 10−20 cm2/Vs

Holes mobility µp 0.0381 cm2/Vs

Metal workfunction φm 5 eV

Table 5.2: Material parameters.


Cross section × thermal velocity σn,pvth,(n,p) 7.8 · 10−19 cm3/s

Density of traps Ntr 1012 cm−2

Trap energy level (acceptor) Etr EV + 0.3 eV

Trap energy level (donor) Etr EC − 0.3 eV

Table 5.3: Trap parameters.

drift diffusion solution (all time derivatives are set to zero).

Time (s) 0 1 2 3 4 5Gate voltage (V) 0 +40 0 -40 0 +40

Table 5.4: Gate sweep.

The purpose of the simulations is to show how the different parameters of a distributionof traps can create macroscopic differences in the shape of the electrical curves. Thesedifferences are mostly given by the variation with time of the threshold voltage, whichis determined by the different amount of charge which is trapped at different times. Atrans-characteristic curve, between different sweeps of voltages, will be translated andmodified in its shape. The same will happen for a capacitance-voltage curve.

5.2.1 Capacitance-Voltage curves

Performing C − V simulations for a MIS structure requires the determination of thedifferential capacitance and must account for the long transient behavior of the traps,which determines the hysteresis. We will use the method explained in [IPR+85], wherein


the differential capacitance is approximated as follows:

C =∂Q

∂V' ∆Q

∆V, (5.26)

where Q is the charge located at the gate electrode plate. In practice the derivativeis approximated with a finite difference and the capacitance is extracted as differencebetween the charges ∆Q in voltage steps of ∆V :

C =Qi+1 −Qi

Vi+1 − Vi

. (5.27)

The method is effective only when applied to an insulated contact, where there only isdisplacement current, and is strictly quasi-static [Lau85].

The capacacitance of the MIS structure in fig. 5.3 will be given by the equivalentcircuit in fig. 5.4 [Sze81]. The insulator capacitance per unit area will be given by

Cit

Cd

Ci

Figure 5.4: Equivalent circuit for the capacitance in a MIS structure.

Ci =εi

di

. (5.28)

The interface traps will give a capacitance

Cit = −∂Qtr

∂Vs

, (5.29)

where Vs is the surface potential. The depletion region, when present with extension W ,will give the depletion capacitance

Cd =εs

W. (5.30)

5.2. SIMULATIONS 69

5.2.2 Simulation results

In this section we show the results for simulations with different parameter sets. In fig.5.5 and 5.6 the curves for the MIS and OTFT structures are shown in the absence of anytrapped state.

Figure 5.7 and 5.8 report the first set of simulations, performed considering differentdensities of traps. Two effects are determined by the variation of the trap density: thereis an increase of the amplitude of the hysteresis and a translation of the curves towardspositive voltages, for increasing trap densities. This becomes clear when one considersthat the traps determine a shift ∆Vth in the threshold voltage, which is

∆Vth = −Qtr(t)

Ci

, (5.31)

where Qtr(t) is the average (along the dimensions of the channel) surface trapped chargeat the instant t.

A change in the amplitude of the hysteresis can also be obtained with a variation ofthe product between the cross section and the thermal velocity, as reported in fig. 5.9and 5.10.

This variation changes the rate at which carriers are emitted and captured: a smallerproduct means higher rates, so the traps are more responsive to the change of bias and,therefore, less hysteresis appears. A difference with the previous simulation is determinedby the absence of threshold shifts, as the total amount of traps that can be filled does notchange.

One issue, which is due to numerical problems, can be observed in fig. 5.9, for thecurve with σvth = 7.8 · 10−18 cm3/s and, in a much lesser way, for the curve with σvth =

7.8·10−19 cm3/s. When reaching the turning point at maximum bias, there is a step, whichis due to an erroneous determination of the charge at the electrode, which is amplifiedby the operation of derivation performed to obtain the capacitance. We were unable tosolve this glitch in the simulations, as reduction of the time steps in the solver was uselessin increasing the accuracy of the results. But we report the curve with the others as itshows, anyway, the expected behavior given by the increase of the product σvth.

The third set of simulations is reported in fig. 5.12 and 5.13. It shows what happenswhen the trap level is located differently in the band gap. For an acceptor trap in a p-type semiconductor the effect of the trap is less when the trap level gets farther from theLUMO. Before a bias is applied at the electrodes, the energy levels are those reported infigure 5.11. The Fermi level is located at a greater energy than the trap level, therefore atthe thermal equilibrium the traps are almost completely filled, giving an overall negativecharge at the interface which shifts the curves on the voltage axis. As soon as a gatevoltage is applied, the gate-induced electric field pushes the electrons away and creates an


−40 −30 −20 −10 0 10 20 30 408.05

8.1

8.15

8.2

8.25

8.3

8.35x 10

−12

Voltage (V)

Cap

acita

nce

(F)

Figure 5.5: C − V simulation for MIS structure with no trap states in the band gap.

−40 −30 −20 −10 0 10 20 30 40−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−6

Vg (V)

Id (

A)

Figure 5.6: Id − Vg simulation for OTFT structure with no traps states in the band gap.

5.2. SIMULATIONS 71

−40 −30 −20 −10 0 10 20 30 408.05

8.1

8.15

8.2

8.25

8.3

8.35x 10

−12

Voltage (V)

Cap

acita

nce

(F)

Nt = 1012 cm−2

Nt = 1013 cm−2

Nt = 1011 cm−2

Figure 5.7: C − V simulation for acceptor traps with densities Ntr = 1011 ÷ 1013 cm−2.Hysteresis is counter-clockwise.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4

5

6

7

8x 10

−6

Vg (V)

Id (

A)

Ntr = 1012 cm−2

Ntr = 1011 cm−2

Ntr = 1013 cm−2

Figure 5.8: Id − Vg simulation for acceptor traps with densities Ntr = 1011 ÷ 1013 cm−2.Hysteresis is counter-clockwise.


−40 −30 −20 −10 0 10 20 30 408

8.05

8.1

8.15

8.2

8.25

8.3x 10

−12

Voltage (V)

Cap

acita

nce

(F)

σ vth =

7.8 ⋅ 10−19 cm3/s

σ vth =

7.8 ⋅ 10−20 cm3/s

σ vth =

7.8 ⋅ 10−18 cm3/s

Figure 5.9: C−V simulation for acceptor traps with σvth = 7.8 ·10−18÷7.8 ·10−20 cm3/s.Hysteresis is counter-clockwise.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4

5

6

7x 10

−6

Vg (V)

Id (

A)

σ ⋅ vth

= 7.8 ⋅ 10−19 cm3/s

σ ⋅ vth

= 7.8 ⋅ 10−20 cm3/s

σ ⋅ vth

= 7.8 ⋅ 10−18 cm3/s

Figure 5.10: Id−Vg simulation for acceptor traps with σvth = 7.8·10−18÷7.8·10−20 cm3/s.Hysteresis is counter-clockwise.

5.2. SIMULATIONS 73

EC

Ei

EFEtr

EF

qs

qm

EV

Vacu

um

le

vel

Figure 5.11: Band diagram of a MIS structure with a trap level at energy Etr

accumulation layer of holes, reducing the overall trapped charge at the interface. This canalso be understood recalling eq. (5.11) and applying some approximations: exchange ofcarriers with the LUMO is rather improbable: that is because the density of free electronsn is negligible and the trap level is located at a far lower level than the intrinsic energylevel. This means that (5.11) simplifies to

dntr

dt' 1

τp

[exp

(Ei − Etr)

kT(1− ftr)− pftr

]. (5.32)

The exponential factor in the right hand side is constant for any applied bias, whereas p

increases as a gate bias is applied. As soon as p becomes dominant, the balance equationreads

dntr

dt' − 1

τp

pftr, (5.33)

which means that the trapped charge will decrease. This also explains why the hysteresisin the curves is always counter-clockwise, as the traps provide negative charge at thebeginning (t = 0, positive threshold shift) and are subsequently neutralized as the appliedbias increases (less threshold shift at the end of the simulation). When the energy levelof the trap is beyond the Fermi level, as in the other simulations of fig. 5.12 and 5.13,the traps are empty, therefore neutral, from the beginning of the simulation, which meansthat there is no threshold shift from the beginning. The applied bias will not change thetrap configuration, and no hysteresis is found.

Simulations in fig. 5.18 and 5.19 show how the hysteresis amplitude can change withthe time length of the sweep, because slower sweeps allow more traps to be charged or


uncharged. This means that a slower sweep will lead to greater hysteresis and thresholdshift.

The same simulations can be performed for donor traps. Figures 5.16 and 5.17 show acomparison between an acceptor trap and a donor trap located at the same energy level.The same device without traps is also reported.

The devices with traps have curves with a different shape, but both are counter-clockwise. The explanation is the same for the acceptor traps: at the beginning of thevoltage sweep the energy level of the traps is located below the Fermi level, therefore thetraps are not occupied by a hole and are neutral. When the gate bias is applied, the trapsare filled with holes and the threshold voltage becomes more negative; furthermore, thehole concentration near the interface reduces, with a subsequent reduction of the draincurrent.

5.3 Summary

In this chapter, we have shown how trap recharging can affect the characteristics of organicthin film transistors. We have described the theory behind charge trap dynamics andhave simulated it for several parameter sets, showing how different threshold voltage andhysteresis amplitudes can be originated. The present work, however, cannot completelydescribe all the phenomena that arise in organic transistors. As a a matter of fact, ithas been reported by Lindner et al. [LPS05], that hysteresis can also be explained bypolarons, whose dynamics might better describe the characteristics of organic transistors.Another possible factor is the presence of mobile ions in some dielectrics, like for examplePoly-vinyl alcohol (PVA), as described by Egginger et al. [EIVS+] or in poly-4-vinylphenol (PVP) [HOH+08]. Since mobile ions are charge which can move when an electricfield is applied, they can change the threshold voltage of the device when they get fartheror nearer the semiconductor/insulator interface. Modeling of mobile ions and polarons isthe necessary future work to get a better insight of the topic. Despite several experimentalstudies already available in literature, a theoretical framework is still needed.

5.3. SUMMARY 75

−40 −30 −20 −10 0 10 20 30 408.05

8.1

8.15

8.2

8.25

8.3

8.35x 10

−12

Voltage (V)

Cap

acita

nce

(F)

Et = E

v + 0.3eV

Et = E

c − 0.3eV

Figure 5.12: C − V simulation for acceptor traps with energy levels at Etr = 0.3 eV + EV

and at Etr = EC − 0.3 eV. The curve with energy level in the middle of the band gapcompletely overlaps with the curve with Etr = EC−0.3 eV. Hysteresis is counter-clockwise.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4

5

6

7x 10

−6

Vg (V)

Id (

A)

Et = E

v + 0.3eV

Et = E

c − 0.3eV

Figure 5.13: Id−Vg simulation for acceptor traps with energy levels at Etr = 0.3 eV +EV

and at Etr = EC − 0.3 eV. The curve with energy level in the middle of the band gapcompletely overlaps with the curve with Etr = EC−0.3 eV Hysteresis is counter-clockwise.


−40 −30 −20 −10 0 10 20 30 408

8.05

8.1

8.15

8.2

8.25

8.3x 10

−12

Voltage (V)

Cap

acita

nce

(F)

S = 400 V/sS = 40 V/sS = 4 V/s

Figure 5.14: C − V simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.Hysteresis is counter-clockwise.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4

5

6

7

8x 10

−6

Vg (V)

Id (

A)

S = 40 V/sS = 4 V/sS = 400 V/s

Figure 5.15: Id − Vg simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.Hysteresis is counter-clockwise.

5.3. SUMMARY 77

−40 −30 −20 −10 0 10 20 30 408.05

8.1

8.15

8.2

8.25

8.3

8.35x 10

−12

Voltage (V)

Cap

acita

nce

(F)

Donor trapAcceptor trapNo trap

Figure 5.16: C − V simulation for acceptor traps, donor traps and no traps. Hysteresis iscounter-clockwise.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4

5

6

7x 10

−6

Vg (V)

Id (

A)

Donor trapNo trapAcceptor trap

Figure 5.17: Id − Vg simulation for acceptor traps, donor traps and no traps. Hysteresisis counter-clockwise.


−40 −30 −20 −10 0 10 20 30 408

8.05

8.1

8.15

8.2

8.25

8.3x 10

−12

Voltage (V)

Cap

acita

nce

(F)

S = 400 V/sS = 40 V/sS = 4 V/s

Figure 5.18: C − V simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.Hysteresis is counter-clockwise.

−40 −30 −20 −10 0 10 20 30 40−1

0

1

2

3

4

5

6

7

8x 10

−6

Vg (V)

Id (

A)

S = 40 V/sS = 4 V/sS = 400 V/s

Figure 5.19: Id − Vg simulation for acceptor traps with sweep rates S = 4 ÷ 400 V/s.Hysteresis is counter-clockwise.

Chapter 6

Conclusion

In this thesis organic thin film transistors have been theoretically studied and modeled,focusing particularly on the electrical characteristics of the devices. We first introduce themodels that describe charge transport in organic semiconductors and current equationsin thin film transistors, focusing on their physical and electrical characteristics.

Then, we develop a model for cylindrical transistors, defining their electrical behaviorin linear and saturation regime, and showing analogies and differences with the theoryfor planar transistors. A comparison with experimental results is also shown. Field-effectmobility is extracted and a heuristic equation for it is proposed, but future investigationsare needed to derive a rigorous analytical approximation. The ultimate goal is to provide acompact model which can be used on simulations in which several transistors are connectedfor circuit applications.

We later analyze organic thin film transistors with short channel effects. More in detail,field-dependent mobility and space charge limited current effects are considered. A modelis proposed for the thin film device, as well as for devices where no electric field exceedsthe boundaries of the semiconductor. Again, a comparison with experimental devices isshown, with good agreement. Future activity on this topic will include a comparativeanalysis with two dimensional drift diffusion simulations, as well as further investigationson the contact effects in the device.

The last part of this thesis addresses the problem of trap recharging effects, which canaffect the characteristics of organic thin film transistors. We have described the theorybehind charge trap dynamics and have simulated it for several parameter sets, providinga critical review of the results. It is worth noting that trap recharging alone cannot modelall the hysteresis effects experimentally measured, so further investigation is necessary, toincorporate other factors like polarons and mobile ions.

On the whole, this thesis has addressed some of the topics that constitute the the-oretical modeling of organic transistors. This research field is rapidly developing as the

79

80 CHAPTER 6. CONCLUSION

properties of the organic semiconductors, which are still partially unknown, are experi-mentally discovered. New device structures are developed, as well as new analytical andnumerical models and theories, so that continuous improvements can enrich both thephysical insight and the engineering in the field of organic electronics.

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List of Publications Related to theThesis

Journal papers

1. Modeling of short channel effects in organic thin-film transistorsS. Locci, M. Morana, E. Orgiu, A. Bonfiglio, P. Lugli - IEEE Transactions onElectron Devices, Vol. 55, no. 10, October 2008.

2. An analytical model for cylindrical thin film transistorsS. Locci, M. Maccioni, E. Orgiu, A. Bonfiglio - IEEE Transactions on ElectronDevices, Vol. 54, no. 9, September 2007.

3. Towards the textile transistor: assembly and characterization of an organic fieldeffect transistor with a cylindrical geometryM. Maccioni, E. Orgiu, P. Cosseddu, S. Locci, A. Bonfiglio - Applied Physics Letters89, 143515:1-3 (2006) (selected for publication on the Virtual Journal of NanoscaleScience and Technology of the American Institute of Physics and the AmericanPhysical Society).

Conference proceedings

1. Investigation on different organic semiconductor/organic dielectric interfaces in pen-tacene based thin-film transistorsEmanuele Orgiu, Mohammad Taki, Beatrice Fraboni, Simone Locci, Annalisa Bon-figlio, Mater. Res. Soc. Symp. Proceedings, Boston, 26-30 November, 2007.

2. Woven Electronics: a new perspective for wearable technologyS. Locci, M. Maccioni, E. Orgiu, A. Bonfiglio, Proceedings of the 29th Annual Inter-national Conference of the IEEE-EMBS, Cit Internationale, Lyon, France, August23-26, 2007.

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90 LIST OF PUBLICATIONS RELATED TO THE THESIS

3. Yarn-like devices for textile electronicsA. Bonfiglio, I. Manunza, M. Maccioni, S. Locci, E. Orgiu, P. Cosseddu, G. LeBlevennec, M. Verilhac, E-MRS 2007 Spring Meeting, Strasbourg, France - May28th-June 1st, 2007.

4. Micro- and nano-technologies for wearable applications in emergencies managementA. Bonfiglio, N. Carbonaro, I. Chartier, C. Chuzel, D. Curone, G. Dudnik, F. Ger-magnoli, D. Hatherall, J. M. Koller, T. Lanier, G. Le Blevennec, S. Locci, G. Loriga,J. Luprano, M. Maccioni, G. Magenes, R. Paradiso, H. Rouault, A. Tognetti, J. M.Verilhac, G. Voirin, R. Waite Proceedings of PHealth 2007 Conference, Porto Carras(Greece), 20-23 June 2007.

5. The textile transistor: a perspective for distributed, wearable networks of sensordevicesM. Maccioni, E. Orgiu, P. Cosseddu, S. Locci, A. Bonfiglio, Proceedings of the 3rdIEEE-EMBS - International Summer School and Symposium on Medical Devicesand Biosensors, MIT, Boston, USA, Sept.4-6, 2006.

Acknowledgements

I remember when I first started to talk about my PhD activity with my advisor, prof.Annalisa Bonfiglio, whom I want to thank for giving me the opportunity to work in herteam: I was so happy, I was about to do scientific research, and even in a fascinatingand attractive topic! I immediately began to study, and soon I made my first scientifichypotheses, I checked their validity, I wrote long, long matlab scripts and equations richin mysterious symbols. Giving meaning to all those strange physical phenomena, gettingnice and useful ideas: yes, that was now my daily challenge. Sometimes hard, but fun,rewarding. I was the only crazy theoretician in my research group in Cagliari, and I wantto thank all of them: Piero, Emanuele, Giorgio, Maurizio, Mauro. Our secretary Simonaand Ileana, also part of my group, deserve a special big GRAZIE!, because they alsobecame very close and true friends.

The usual PhD routine went on for two years, then the lightning. I got the chanceto join a new research group, far away, but I was a bit scared, having to change my lifecompletely from one day to the other. I accepted, and the new home for me became:München! Warm temperatures and crystalline sea, joy and delight of Sardinia? No more!The north icy wind, the soft white snow, suddenly a revolution. I came to München alone,knowing nobody and ignoring completely even the meaning of the simplest words: GrüßGott, tchüß!, schnitzel :) . The adventure was just starting. I met there my new researchgroup and I could immediately notice a difference with my group in Cagliari: there, I wasthe only theoretician, being the others all experimental guys; now all of us were, let’s say,math lovers.

Now I want to thank them: of course prof. Paolo Lugli, my new boss, our secretariesBettina and Rita, Giuseppe, Christian. But also the two other Italian Phd students ofthe group, Mario and Federico, with whom I had so much fun, starting from rememberingstrange sentences like “Solo Puffin ti darà, forza e grinta a volontà!” (I think in German it’s“Nur Puffin schenkt mir die Kraft und Ausdauer die ich brauche”, in English I don’t knowthe exact sentence, but ok, somebody knows what we were remembering :)), continuingwith repeating countless times and laughing at the famous press conference, in 1998, ofthe best footlball trainer ever, Giovanni Trapattoni (Ich habe fertig!!!), and ending with

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92 ACKNOWLEDGEMENTS

terrible falls, at least for me, with the snowboard on the Austrian Alps. And part of thisbunch of crazy friends is also Alpar, with whom I could see the beauty of his country,Rumenia, and also had lots of fun. Still part of my group are Omar, Alee, Daniela,Sebastian, Stefan, Christoph and Rosi. Thank everybody.

I met another person here in München. Among the other things, she taught me, withinfinite patience, a bit of German language, and helped me understand the wonderfulculture of Germany. She always has a kind smile for me, so special GRAZIE for you,Nicole!

Of course I did not forget my good old friends. Whenever I was going back to Sardinia,I always enjoyed partying with them. Claudia, Greca, the boys and girls from the gym;Alessandro deserves again a special GRAZIE, being my best training partner and also abrother for me.

Could I not mention the lady that also started a PhD with me, and with whom I sharedjoys and sorrows of the PhD? We laughed together at reading PhD comics, because theyare so dramatically true. So thank you, Alessandra. And with her, I also want to thankall my good friends met at the university, some of them are around the world now, butactually they are near: Nicola, Antonio, Carlo, Frichu, Bruno.

And last, but not least, I would like to thank my mother, my father and my sister, orshould I say Ire, Nico and Cri? I always call them by name, strange. They have alwaysbeen with me. The last in my long list of thanks goes to Nonna Nina, who had just a fewproblems recently, but now smiles again.

So, I am at the end of the story. What’s next is kindly unknown, and still has to bewritten.

Simone

Modeling of Physical and Electrical Characteristics of ... · Modeling of Physical and Electrical Characteristics of Organic Thin Film Transistors ... Dept. of Electrical and Electronic

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